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OF CONVEX BODIES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By Antonis Tsolomitis, B.Sc.
****
The Ohio State University
1996
Dissertation Committee: Approved by W. Davis, Advisor
G. Edgar / / Advisor V. Milman Department of Mathematics M. Talagrand UMI Number: 9630994
UMI Microform 9630994 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 to my parents who gave me life and to my teachers who taught me how to live it
ii AKN OWLEDGEMENTS
I express deep appreciation to Professor V. Milman not only for his guidance and insight throughout the research but also for showing me the philosophical aspects of Mathematics through Wittgenstein’s assertion: Mathematics is a lan guage ([27]). Gratitude is expressed to Professor W. Davis for without him this dissertation would not have been possible. Thanks go to the other members of my advisory committee, Professors G. Edgar and M. Talagrand, for their suggestions and comments. I offer sincere thanks to Professor W. Johnson for making my visit to Texas A&M University possible. The technical assistance of Dr. G. Bhatnagar,
V. Gougoulides and P. Zafiropoulos is gratefully acknowledged. I offer sincere thanks to the communities of the Hellenic Students at the Ohio State University and Texas A&M University for their support and especially to E. Mavromatidou,
Dr. C. Rouvas and Z. Zeniou. Special thanks go to A. Apostolopoulos and E.
Mavromatidou for the philosophical discussions we frequently had. To my family in Greece, I thank you for your transatlantic support and humor. VITA
June 25, 1967 ...... Born, Piraeus, Greece
1989 ...... B.Sc., University of Athens, Athens, Greece
1989-1995 ...... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio
1995-1996 ...... Presidential Fellow, The Ohio State University, Columbus, Ohio
FIELDS OF STUDY
Major Field : Mathematics
iv TABLE OF CONTENTS
DEDICATION ...... ii
ACKNOWLEDGMENTS ...... iii
VITA ...... iv
TABLE OF CONTENTS ...... v
LIST OF FIGURES...... vi
CHAPTER
I. Introduction ...... 1
II. B ackground ...... 12
III. Symmetrizations O f C onvex B odies ...... 27
IV. C onvolution O fC onvex B odies ...... 56
REFERENCES...... 92 LIST OF FIGURES
FIGURES
1. Lower bound of the symmetrized body K ...... 50
2. Lower bound of v.rad.(K OF) ...... 54
vi CHAPTER I
INTRODUCTION
1 .Prolegomena
Geometry is one of those elements of human civilization whose value can not be overestimated. Looking back in the history of science we find geometry to be its very foundation. Starting from the need to “measure the earth”-as its name reveals (geometry < 'yea + tierpu = earth + to measure)~it was the first discipline to become into a rigorous science. It was through Geometry and Philosophy that the purely abstract thinking was developed and the ideas of theory and proof were born. Consequently geometry is in the basis of practically everything. From the construction of the musical scales by Pythagoras, to the way we understand our universe after its geometric description by Aristarchus. From something as theoretical as physics to something as applied as medicine. Thus it would not be an exaggeration to say that Geometry is one of the fundamental elements of the history of human civilization.
1 One of the problems that appeared in early geometry was what is today called
“the isoperimetric problem”. The problem asks to find that closed plane curve of given length, that encloses the maximum possible volume and the circle is the solution. The problem made its appearance in antiquity and information about arguments to its favor have reached us through the work of Zenodor (2nd century
B.C.; see [5]). The first rigorous proof for the circle being the solution of this problem, was given in the year 1882 by Edler.
It is not difficult to see that if the maximum of enclosed area is achieved by some curve, then the enclosed region must be a convex set, i.e. the intersection of this region with any line of the plane consists only of one line segment. One of the ways to verify the solution of the isoperimetric problem is to use the Steiner symmetrization. To symmetrize a segment of finite length on the plane with respect to a perpendicular to the segment line, means to move the segment in the direction vertical to the line until it is bisected by the line. Let K be a region in the plane enclosed by a curve so that it is convex. Let £ be a line in the plane.
Symmetrizing with respect to I the intersections of K with all lines perpendicular to £, we arrive at a new body K , the Steiner symmetrization of K. This procedure preserves the volume but increases the length of the boundary of K. Moreover we can keep symmetrizing with respect to certain lines, so that the resulting bodies
“converge” to a Euclidean ball. The sense of this convergence will be defined rigorously in the next chapter.
Thus if we want to keep the length of the boundary fixed, the volume must be increased and since the symmetrizations lead to a Euclidean ball, that must be the solution to the isoperimetric problem.
Passing now to higher dimensional spaces and understanding their geometry is something that science was and still is in need for. The need of many dimen sions does not stem from mathematical curiosity only (although this would be considered by many philosophers and mathematicians a complete justification of the need) but also from the needs of applied science and theoretical physics. For example the mathematical basis of quantum mechanics is the theory of linear op erators in a Hilbert space. The success of Hilbert spaces in such applications lies on the fact that their geometry is “essentially Euclidean” and hence we have a very good understanding of them. Since non-Hilbert spaces appear naturally in applications it seems necessary to study the geometry of these spaces.
The difficulties that appear in this study come from the existence of many counter-intuitive phenomena that take place in “high” dimensional spaces. One of these, in the language developed by V.D.Milman, is the “concentration of measure phenomena”: let e > be given. Cut out from the n-dimensional Euclidean sphere in Rn two cups around the north and south poles of the sphere using two parallel hyperplanes being at distance e. What remains is a “band” around the equator of height e. The surface area of this band converges exponentially fast to the surface area of the entire sphere as the dimension n increases to infinity (one should normalize the measure on the sphere so that its surface area equals 1).
This means that in high dimension the surface measure on a sphere concentrates strongly around its equator (any equator).
The isoperimetric problem in many dimensions (asking for maximal volume enclosed by a surface of given area) as well as on the sphere (asking for maximal surface area on the sphere enclosed by an n —2-dimensional manifold on the sphere) is again verified using symmetrizations. Analogues of Steiner symmetrizations are one part of this dissertation. Precise definitions are given in the next chapter, as well as the presentation of relevant material from the Local Theory of Banach
Spaces and Convexity Theory.
Another part of this dissertation is the study of the convolution body of two convex bodies in Kn. This is connected with another phenomenon of high dimen sion, namely the instability of volume. For an example of this instability let us consider the Euclidean ball of radius 1 and normalize the volume element in RTl 5 so that its volume is equal to 1. Then considering a smaller ball inside the last one of radius (1 — e) (for a small but fixed e) its volume is (1 — e)n, converging very fast to zero as n increases to infinity. This means that with the dimension increasing, the Euclidean balls become more and more “hollow” and their volume concentrates strongly in a neighborhood of their boundary.
If K and L are convex bodies in Rn, their convolution body of parameter
0 < 6 < 1 is constructed by all vectors in Rn so that when L is shifted by any of them, the common volume of K and the shifted L is more than tivo^if n L).
Here we identify these bodies-up to homothety-for S tending to 1.
The classical Euclidean geometry did not have any essential progress after the
“death” of the Alexandrian School under the famous woman philosopher and mathematician Hypatia (she was murdered by a fanatic Christian crowd the year
415 A.D. minutes before the burning of the second big library of Alexandria,
Egypt, by the same crowd). In the Renaissance it was “algebrized” by Descartes simplifying on the one hand many proofs but taking out most of its beauty on the other; a brutal but useful action as it was characterized by many historians.
Modern Convexity theory started with the works of Cauchy and Steiner in the
1840, and continued with the work of Brunn, Minkowski, and later with Alexan 6 drov and Fenchel. In the second half of the 20th century the Local Theory of Ba- nach Spaces started developing with the works of Grothendieck, Dvoretzky, and later with the works of Vitali Milman, Gilles Pisier, Jean Bourgain and others.
Both theories could be viewed as being closely connected with classical geome try on the ideological level, although there are direct connections through, for example, the isoperimetric problem. A very interesting observation is that the notion of derivative plays not such an important role here as it did not play any role in classical geometry (since of course it did not exist). The study of volume and consequently the integration is central here as it was in antiquity. Moreover substructures are examined, projections, transformation of objects into new ones with certain properties, orthogonality properties are used and so on. In the next we develop the main points of Local Theory of Banach Spaces and Convexity
Theory, that either relate to the results of the dissertation or help to develop a more solid basis of results in the direction of these theories.
2.The general context
The local theory of Banach spaces started in the second part of the century.
The first result of the theory could be considered to be the Dvoretzky’s theorem, and most of the results that followed were shaped around it. Dvoretzky’s theorem was strengthened by V.D.Milman. From the philosophical point of view though, 7 the contribution of the proof of V.D.Milman lies on the fact that it gave us the ability to understand much deeper the high dimensional spaces and develop the
“right” intuition, exploiting the concentration of measure phenomena. Let us state the Dvoretzky’s theorem in the form proved by V.D.Milman (see [7] or [20]).
T heorem 1.2.1. (Dvoretzky’s) There exist a universal constant c so that for any n-dimensional normed space X = (Rn , | | . ||) and any e > 0 there exists a subspace F of Rn so that
(1) dim F > ce 2 logn
(2) for all x in F , M ( 1 — e)|x| < ||x|| < M( 1 + e)|x| where | • | denotes the norm of the standard Euclidean structure in Mn and M a constant (the median of the norm).
Actually, we have used an improved form of this theorem as far as the depen dence of dimF on e is being concerned (see [8] or [24]).
The proof is based on the following fact proved by P.Levy in 1919 ( in a modern understanding this fact it is called the concentration of measure phenomena for
S n_1):
For any set A in §n_1 and e > 0 we denote A e — {x; p(x, A) < e}, where p is the geodetic distance on S"-1. 8
The next theorem describes an isoperimetric property of the sphere:
T h eorem 1.2.2. For each 0 < a < 1 and e > 0, min{/i(Ae); A C
§n_1, p(A) = a} exists and it is attained on A 0; a “cap” of measure equal to a.
By a “cap” we mean a set Ce for C C 8n 1 being a singleton and e > 0.
If we use the above theorem for a = \ (in which case A 0 is a hemisphere) we can compute that there is constant c so that for A C §n_1 with p{A) > | we have,
(1.2.3) tx(Ae) > l - c e - e’n'2.
Let now / be a continuous function on Sn_1, and u>f(e) its modulus of con tinuity. Let Mf be the median of / (Levy mean), i.e., Mf is that number that satisfies both
p{x e Sn_1 ; f(x) < Mf } > i and p{x G Sn_1 ; f{x) > Mf] >
Thus we conclude the Lemma of Levy i.e., if A = {x ; f(x) = Mf} then
(1.2.4) p(Ae) > 1 - ce"e2n/2, since,
(1.2.5) Ae = {xe S " -1; f{x) < Mf }e f]{o: G §n~ \ f{x) > Mf }e. 9
This consequence says that if the modulus of continuity of / satisfies u>f{e) < 5 then | f(x) — M f \ < 8 on Ae which is of “big” measure. This means that a well behaved function is “almost” constant on “almost” all the sphere. The Dvoretzky theorem’s proof is based on this observation which is used with the arbitrary norm
||. || in the place of the function /. The modulus of continuity though of ||. || turns out that requires a lower estimate of the median The estimate
(1.2.6) % l l > c ( ^ ) 2 is proved using the classical Dvoretzky-Rogers Lemma (see [20]):
Lemma 1.2.7. Let X = (Rn , ||. ||) be an n-dimensional normed space and let
D be the ellipsoid of maximal volume inscribed in the unit ball Bx of the space
X . Then there exists a basis (xj)™=1, which is orthonormal with respect to D such that
(1.2.8) 2 < ||x|| < 1, for j = 1,2,..., n — 1.
Of course such a deep understanding of high dimension was bound to produce many more impressive results. Thus we have the inverse Brunn-Minkowski in equality [18], the inverse Blaschke-Santalo inequality [4], the subspace of quotient theorem [17], etc. . In Convexity theory the Brunn-Minkowski inequality is of central importance (see [21] or [26]): 10
THEOREM 1.2.9. (Brunn-Minkowski). For convex bodies K ,L in Rn and for
0 < A < 1,
(1.2.10) vol(( 1 - A)K + AL)1/n > (1 - A)vol(K )^n + Avol(L )^n.
Equality for some A e (0,1) holds if and only if K and L either lie in parallel hyperplanes or are homothetic.
The converse of it even up to constant fails trivially. It is possible though to get an inverse of Brunn-Minkowski if the bodies involved are first put in a “good” position. The next is due to V.D.Milman (see [18]):
T heorem 1.2.11. (Inverse Brunn-Minkowski). Two convex symmetric bodies
K and L in Rn can always be transformed (by a volume preserving linear isomor phism) into convex symmetric bodies K and L which satisfy,
(1.2.12) vol{K + L )1/n < C (vol(K )1/n + uo/(L)1/n) , where C is a numerical constant independent on n. Moreover, the polars K° and
L° and all their multiples satisfy a similar inequality.
This is one of the very important connections of Convexity and Local Theory.
Similarly in Convexity we have the Blaschke-Santalo inequality [23], i.e., for every n > 1 every convex symmetric body K in Rn satisfies, P S g P ) ‘s'. and from Local Theory the inverse Blaschke-Santalo inequality due to J.Bourgain and V.D.Milman [4], i.e., there exists universal constant C > 0 so that for any n > 1 every convex symmetric body K in Rn satisfies,
, ( vol{K)vol(K°)\ (1.2.14) ~ V vol(Bq )2 J
The proofs are based on the quotient of subspace theorem due to V.D.Milman
(see [17]):
THEOREM 1.2.15. (Quotient of subspace theorem). Let X be an a n-dimensional norrned space and let X < 1. There exists a quotient space Y of a subspace of X with dimY > Xn and
(1.2.16) d ( Y ,4 >mY) < c(l — A)_2|log(l — A)t, where c is an absolute constant.
This theorem is using an iteration argument and the very non trivial fact that if | | . || is the norm of X , then there exists a Euclidean norm | . | on X for which
M||.||M||.||. < clog(rf(Jt, ^2)) f°r some absolute constant c (and a -1 \x\ < ||a:|| < b\x\ with a/M||,||. < \/n). The existence of such a Euclidean structure as described above is based on the work of G.Pisier, T.Fiegel/N.Tomczak-Jaegermann and
D.Lewis (see [20]). The Local Theory of Banach Spaces has also other important branches. One of them is the theory of type and cotype developed by Bernard Maurey and Gilles
Pisier (see [21]). G.Pisier is recently developing another direction on the operators level. We will not attem pt to go into these directions here. CHAPTER II
BACKGROUND
1. Sym m etrizations
In order to measure the distance between two convex bodies K and L in Rn , we use the Hausdorff distance defined by,
5(K,L) = inf{a6 : a~lL CifC bL}.
We will also use the notation v.rad.(K) for the quantity (vol(K)/vol(Bn))l/>n, where Bn stands for the Euclidean ball of radius one.
Let K be a convex symmetric body in Rn and let a G Sn_1. The Steiner symmetrization of the body K with respect to the vector a is the body
au{K) = {x + Xu : x G P[U]±(K) and |A| < v.rad.{{x + [a]) H K)}, where [a] is the subspace spanned by a and the orthogonal projection on the orthogonal complement [u]1 of [a].
The Minkowski symmetrization with respect to a is the body
r . m = i (K + *v(K )),
13 14 where 7ru : Rn —> Rri is the reflection through the hyperplane [u]1- and the addition “+ ” denotes the Minkowski addition (A + B = {x+y : x € A and y e B} for A, B subsets of R").
If is a sequence of vectors in Sn_1 then we will write
n IK* i= 1 for the set
a u n (^U n-i (• • ' a U 2 (ffU i(K )))) •
Similarly for n?=i ru,K.
Our starting point is the following theorem of Bourgain, Lindenstrauss and
Milman (see [2] ):
T heorem 2.1.1. There exist absolute constants co,c so that if K is such that v.rad.(K) = 1, there is an ordered sequence (ui)^=1 of unit vectors in Rn with k < con log n, so that if the Steiner symmetrizations with respect to this sequence are applied with this order to K , will give a body K satisfying
(2.1.2) c~lB n C K C cBn.
If £ > 0, then with k < c(e)n more Steiner symmetrizations, we get a body K satisfying
(2.1.3) c~lB n C K C {l + e)Bn. 15
This theorem was proved using many techniques of the local theory of Banach spaces and several things known about Minkowski symmetrizations from [1], whose main theorem is,
Theorem 2.1.4. Ife > 0 andn > no(e) and if we perform N = cn log n+c(e)n random Minkowski symmetrizations, then with probability at least 1 — exp(—c(e)n) we get a body K such that
(2.1.5) (1 - e)rBn C K ( 1 + e)rB n, for some r = r{K), constants c, c(e), c(e) and no(e) positive integer.
By “random Minkowski symmetrization” we mean the procedure in which the direction of the symmetrization is chosen randomly on S n_1 with respect to the rotation invariant measure of S"-1 .
The dependence of the c(e) appearing in the previous statement on e is c(e) « exp(ae| loge|) where here, a is a universal constant.
The dependence of N on n is optimal as the following example from [1] shows:
Let e € Sn_1 and consider the norm ||a:|| = | < e ,x > \ + \x\/y/n. For this norm
M||.|| « n-1/2. After k random Minkowski symmetrizations we get a norm whose expected value at x is « (l — jj-)* | < e, x > \ + \x\/\/n. Thus unless k is bigger than Inlogn we get a norm whose expected value for e is much larger than its average value M||.||. 16
2 R emark 2.1.6. The no(e ) behaves like ) , hence no result is stated in theorem (2.1.4) for n < no(e). Since theorem (2.1.4) is used in the proof of the upper e-estimate in theorem (2.1.1) no result is stated there as well for n < n0(e).
The proof of the e-estimate from above in theorem (2.1.1) is essentially based on the fact that for every u € S "-1 the following holds:
(2.1.7) aU(K) C tu(K) and thus the above e-estimate for Minkowski symmetrizations can be used. Since the converse of the last inclusion fails in a bad way the below e-estimate for
Minkowski symmetrizations can not be used to provide the e-estimate from below for the Steiner symmetrizations. It is not yet known how to get this e-estimate from below using a “small” number of Steiner symmetrizations. However it is clear from the proof given in [2] that it would be possible to get this estimate if it was possible the moment that Mk° is sufficiently close to m ax^^n-i ||a7||ic°, to find a subspace H of of dimension > (1 — e)n so that on H, K is e close to a
Euclidean ball. But this can not be ensured in general.
As far as the proof of the first part of theorem (2.1.1) is concerned (i.e. the fact that very few Steiner symmetrizations are needed in order to bring the body K to a fixed distance from a Euclidean ball), the following fact taken from the proof of theorem (2.1.4) together with Dvoretzky’s theorem and the fact that there are 17
less than n Steiner symmetrizations that can change an ellipsoid to the Euclidean
ball of equal volume play the central role:
Fact 2.1.8. Assume that
(2.1.9) a-'B„ C K C 0 Bn
and that
J S"-1 then there exist (uj)j=1 with k < cn (c an absolute constant ) so that k k (2.1.11) \ [ c u K C Y [ r u K C j = 1 j =1
From the proof of theorem (2.1.1) we will need for our main results the following lemma:
Lemma 2.1.12. Let 0 < Ao < 1, 0 < 7 < 1 and let Bn denote the Euclidean
unit ball in Rn . Let H be a linear subspace ofW1 of dimension h > 7n. Then the
convex hull of AoB n U (B n fl H) contains an ellipsoid £ such that
v.rad.{£) > (77(1 - 7 )1 7) 1//2 Aj y .
Turning now to theorem (2.1.4), it is proved in its dual form since it is easier to work with norms. The dual statement is as follows (the norm || . || in this statement corresponds to the norm of the polar body of K , appearing in theorem
(2.1.4)): 18
T h eo r em 2.1.4'. Zet||.|| be a norm in W 1 and let N = c n lo gn+ c(e)n (where e > 0 and n > no(e) ). Then for all (upfL x in (S71-1)^ with an exception of a set of (crn)N measure at most exp (-c(e)n) we have
(2.1.13) (1 - e)r|x| < 2~N ^ || 7rUix|| < (1 + e)r\x\, i 6 Rn , D i£D where the sum is taken over all subsets D of { 1,2,..., N } and
(2.1.14) r = f \\x\\dan(x)= [ \\Ux\\dpn(U). J S" - 1 Jo n
(nu denotes the reflection operator with respect to the hyperplane perpendicular to u).
Its proof is based on the fact (2.1.8) and on the following:
FACT 2.1.15. If is more than p/8, then for all e > 0 and n > n(e) and with N = c(e)n
(2.1.16) (1 - e ) M |t.|||®| < 2 _iV^ | | rrUix\\ < (1 + e)Af||.|||s|, x e Rn , D i£D with probability of at least 1 — exp(— c(e)n), where the sum is taken over all subsets of {1,2,..., N}. c(e) is a constant depending only on e.
Both facts (2.1.8) and (2.1.15) use an application of the concentration of mea sure phenomena on the orthogonal group On. 19
These theorems were the motivation to start investigating higher dimensional symmetrizations. We continue now by presenting more results that will be used in proving our main theorems.
The following lemma from [4] is a consequence of a classical result
(see [22] ):
L em m a 2.1.17. Let K be a convex symmetric body in R” and F ^ Rn of dimension k. Then
M b-n+ k f ^ vol(KnF) < vol(K) < an~k ( C,nZt } vol(KnF) \kj \y/n — kj \ \ / n - k J
where a, 6 6 R are such that a~ 1B n C K C bBn and R 3 cn 1 c > 0 is such n-too that vol(Bn)" =
Next is a consequence of Busemann's formula. This formula states that if
K \, K 21 ■ • •, K n- i are convex bodies in R” , then
n — 1 « (2.1.18) Ylvol(Kj) = n\vol(Bn) T{KX n H ,..., K n- i n H)dH, j = l ''Gn.n-l where
T(K inH ,K 2 nH ,...,K n- 1 nH) =
= ••• / T(0, x n- i ) d x i ... dxn- i J K xnH 20 and T(0 , x \,..., a;n_i) denotes the volume of the simplex with vertices 0, xi,...
. ..;cn_i, thus T(0,xi,... ,rc„_i) = ^iji|dei(Ei,... xi,...,x n- i e H.
Gn,n—i is the grassmanian with indices n ,n — 1 and dH the normalized Haar measure on it.
Prom this, the following proposition follows (see [19] ):
Proposition 2.1.19. Let K be a convex body in Mn . Then
(vol(lC))”- 1 = n\vol(Bn) ( (volZ(K n H)) (vol(K n H ^ ^ d H
where Z (K D H) is the zonoid of inertia of K C\H.
The following proposition is due to Busemann (see [ 6] ):
Proposition 2.1.20. Let K be a convex symmetric body in Kn . Then
(2.1.21) v.rad. (Z(K)) > v.rad.(K).
From propositions (2.1.19) and (2.1.20) we get easily (see [6] ) the following fact:
Fact 2.1.22. If K is a convex body such that for any hyperplane H through the origin
(2.1.23) vol(K C\H) > vol(Bn C\H) — vol{Bn- 1) 21 then
(2.1.24) vol(K) > vol(Bn).
R e m a r k 2.1.25. It is unknown if B n can be substituted in the previous fact with any convex symmetric body K\ even up to some universal constant. I.e. it is not known if there exists absolute constant c so that whenever K,K i satisfy
(2.1.26) vol(K i n H) < vol(K n H), it will be implied that
(2.1.27) voZ(.K’i) < c vol(K).
In [19] it is shown that this problem is equivalent to the hyperplane conjecture.
Many other equivalent statements are presented in this paper. 22
2. Convolution bodies
Let us start by defining the standard convolution body:
D efinition 2.2.1. For 0 < 8 < 1, the convolution body of parameter 8 of the convex symmetric body K is the set
(2.2.2) C(S; K) = {x 6 Rn : vol n(K n (a: + K)) > 8vo\n(K )}.
When it is clear to which body K we refer, we will write C( 8) om itting K.
The convolution body C( 8) is symmetric (since K is), and it is a consequence of
Brunn-Minkowski inequality that it is convex:
vol n{K D (Ax + (1 - A)y + K))1/n =
= volri((Aii: + (1 - A)K) n (A(® + K) + (1 - A)(y + K)))l>n
> voln ((A K fl A (a; + K)) + ((1 - A )K n (1 - A)(y + K)))1/n
> voln (Kn(x + K))1/n + (1 - A)vol„ (K n {y + K))l/n .
K.Kiener proved the following (see [11] ):
T h eorem 2.2.3. Let K be a convex symmetric body in lEP with voln{K) = 1
and 0 < 8 < 1 . Then
(2.2.4) lim = voU P l), 23 where is the polar of the projection body Pk of K .
This theorem is based on a direct argument and an application of Fubini’s theorem.
Later on, M Schmuckenschlager strengthened the above result, proving the following (see [25] ):
T heorem 2.2.5. Let K be a convex symmetric body in Rn with vol(K) = 1 and 0 < S < 1. Then
(2.2.6 ) (1 - S)P°K C C(S) C lo g (i)/$ .
In particular
(2.2.7) where P£ is as in the previous theorem.
Schmuckenschlager’s argument uses an exponential bound on the convolution square that resembles concentration of measure techniques.
The convolution square of a compact convex subset C of Rn is the function
F = xc * Xc where xc is the characteristic function of C and * denotes the convolution operation. It is readily verified that if C = K is a convex symmetric body then for t > 0 and u € § n_1 we have that F(tu) = vol n(K n (tu + K )). 24
It is very easy to get a lower bound for the convolution square of K and theorem
(2.2.5) follows by getting a suitable upper bound for F.
The formula in the last theorem gives easily information about the projection body of K. For example although it is not clear a priori that the projection body satisfies:
(2.2.8) PnK )=T*(PK), for every invertible transformation T, one can see it very easily through this
formula, since C(6 ;T(K)) = T(C(S;K)) for such transformations:
[C(<5;T(lf)) = { r 6 R n : voln((a: + T(K)) fl T(K)) > tSvol^IX.K'))} =
= { i e f : |det(T)|voln((T" 1a: + K) D K) > 5\det{T)\vo\n{K)} =
= T({x G P : vo\n((x + K) D K) > 5voln(X)}) =
= T(C(6 ; if))]
For more information of the projection body one may see the recent survey [3] or the book [26].
We consider now the convolution body of two different bodies:
Definition 2.2.2.9. Let K and L be two convex symmetric bodies and 0 <
S < 1. We define the convolution body of K and L of parameter S to be the set
(2.2.10) C{5-,K,L) = {xe Rn : vo\n(K D (x + L)) > Svo\n(K D L)}. 25
It is obvious that C(6 ;K,L) is not affected if K and L are interchanged in the definition. C(S\K,L) is symmetric and it is again a consequence of Brunn-
Minkowski inequality that this set is convex (a similar argument like the one presented for the case with L = K above, works also here). It is also immediate
from the definition that if 8 —> 1 then C(S\ K .L) “shrinks” down to zero, provided that voln (K D (a: + L )) < vo\n(K H L) for all x € Rn \ {0}.
We will look for the “right” normalization exponent a, such that the limit:
lim <5-H (1 - 5)“ exists (in the Hausdorff distance), and does not degenerate. We will call this limit- body the “limiting convolution body” of K and L, and denote it with C(K,L).
By “degenerate set” we mean a set with empty interior. The case of an infinite cylinder is considered to be a non-degenerate case and it may happen to be even
Rn. We understand convergence as convergence (in the Hausdorff distance) of the intersections of our sets with any (fixed) Euclidean ball in R. There are examples where for 0 < ao < 1/ 2, the limiting convolution body collapses to a point for all exponents a < a 0 and it converges to an infinite cylinder or R, for all a > «o.
In particular theorem (2.2.5) says that:
(2.2.11) C(K,K) = P°. 26
The body C(K,L) when it exists, is also affine invariant since it is easy to see that if T is any invertible linear transformation of Rn then,
C(T(K),T(L)) = T(C(K,L)).
[since C(5;T(K),T(L)) =
= {x € K” : vo\n(T(K)n(x + T(L)))> 6 vo\n{T(K)nT(L))} =
= {i £ f : |det(T)|voln(Ar n (T~*x + L )) > 8 \det{T)\vo\n(K fl L)} =
= T(C(8 ;K,L))]. CHAPTER III
QUANTITATIVE STEINER/SCHWARZ-TYPE SYMMETRIZATIONS
1. fc-syramet r izat ions
Let us first recall the definition:
D efinition 3.1.1. For all k e N so that 0 < k < n the ft-symmetrization of the body K with respect to the /^-dimensional subspace F of Rn , is the operation that changes K to K with
K = {x + Au : x E Ppx(K), u E F, |it| = 1, |A| < v.rad.((x + F ) f l K )} where Ppx (K) is the orthogonal projection of K on F 1 and | . | is the standard
Euclidean norm.
Observe that if k = 1 then we have the Steiner symmetrization and if k = n — 1 the Schwarz symmetrization. By Fubini’s theorem it is clear that this sym metrization preserves volume. Moreover, these symmetrizations preserve con vexity, and they change ellipsoids to ellipsoids. One can easily see this through
Steiner symmetrizations since those have these properties (see [2]), and every fc-symmetrization is the limit of a sequence of Steiner symmetrizations.
27 An-symmetrizations, A = £ < 1/2, k € N.
In this case, consider any set of [^] pairwise orthogonal subspaces of dimension
An (here [.] denotes the integer part function). Symmetrize the body with respect to all of those subspaces plus an additional subspace that contains the remaining dimensions (this last subspace is not necessarily orthogonal to the previous ones).
It is readily verified that after these symmetrizations the body K has changed to a body, say K i, which is close to an ellipsoid £ and satisfies
(3.1.2) and consequently,
(3.1.3)
We now see how to symmetrize this ellipsoid to a Euclidean ball:
PROPOSITION 3.1.4. Let £ be an ellipsoid, with v.rad.(£) = 1. Then there are at most ([l/A] + 1) An -symmetrizations that will change £ to Bn. P r o o f .
Claim. For every 0 < k < n natural number, there exists subspace F of Rn of n X? dimension k, such that v.rad.(Fn£) = 1. [Assume £ = {x = ^ 3 = 1 j 1}. Since v.rad.(F C\£) = 1, we conclude that
n (3.1.5) n ° j = i - 3 = 1
Without loss of generality we may assume that oi > 02 > • • • > an, hence with the aid of (3.1.5) we get that 0102 ... a*, > ana„_ 1 ... an-k+ i■ We conclude that
there are subspaces F\, F2 of En with dimFi = k (i = 1,2) so that
(3.1.6) v.rad.(Fi D £) < 1 < v.rad.(F 2 fl £).
By using the intermediate value theorem on the (path connected) Grassmanian
Gn,k we get that there is F with dim F = k and v.rad.(F D £) = 1]
Now let Fi be a subspace of Rn with dim.Fi = k and v.rad.{F\C \£) = 1. Let £\ be the result of symmetrizing £ with respect to Fi. If fci, 62, - • -, bn are the lengths of the semi-axis of £\ by the choice of Fi and (3.1.5) we have (after renumbering if necessary):
(3.1.7) bi — 62 = • • -b\n = 1 = 5,\n+l^An+2 • • * 30
We now distinguish between the following two cases:
Case 1. X >
Take a subspace F 2 of Rn so that F t, dim F2 = Xn and symmetrize with respect to F2. Let £% be the resulting ellipsoid. By the choice of F 2 and the symmetrization we have
(3.1.8) F2 H £2 = 5 (X_A)n and
(3.1.9) F 2 n £2 = R B \n, for some R > 0. Since ellipsoids and volume are preserved under these sym metrizations we must have R — 1 and thus £2 = B n
Case 2. X <
In this case we take a subspace F 2 of R” so that Fi F ^ and such that dim F2 = Xn and u.rad.(F 2 n£i) = 1.
Symmetrizing with respect to F 2 we get an ellipsoid, say £2, with lengths of semi-axis (cj )”=1 satisfying
(3.1.10) Cl = C2 = • • • = C\n — C \n + \ — • — C2Xn — 1 — 1 1 Cj j>2An
(note that c* = bi for all i = 1,2,...,An). Choose now a subspace F 3 of Rn
with dimF3 = An,v.rad.(F 3 fl £2) = 1 and Fi U F2 C Fj-. Symmetrize with 31
respect to F3 and continue in this manner. If 1/A is not an integer then the last subspace with respect to which we symmetrize is any subspace that contains the orthogonal complement of the span of the previous [1 / A] subspaces. Hence after at most [^] + 1 symmetrizations the length of all semi-axis will have become equal to 1 □
Thus we have arrived at the following:
T heorem 3.1.11. For every convex symmetric body K in Rn and any 0 < A <
1 such that Xn is an integer, there are less than 2 ([y] + l) Xji-symmetrizations that will change K to a new body K satisfying:
(3.1.12) c~xrBn C K C crBn, where r = v.rad.(K) and c = [^] + 1 . □
Arc-symmetrizations, A = £ > 1/2, k € N.
We start with the following definition:
Definition 3.1.13. We call an “orthogonal pair of An-symmetrizations” a pair of symmetrizations with respect to subspaces F and H of Rn , where dim F = dim H — Xn and H 1- is a subspace of F. 32
W ith this definition we have the following theorem:
T h e o r e m 3.1.14. Let K be a convex symmetric body with v.rad.(K) = 1 sat isfying:
(3.1.15) a~lBn C K C bB n, with a, b real numbers such that m ax(a, b) < \/n. Let X = £ with k € N such that
| < A < 1. Then we have
(1) There are two orthogonal pairs of Xn-symmetrizations that will change K
to K satisfying:
c~lBn C K C cBn
with c being an absolute positive constant independent of X and n.
(2) For | < A < 1, A N Y C'loglog(n) orthogonal pairs of Xn-symmetriza
tions, will change K to K satisfying:
c~ 1Bn C K C cBn
where C and c are absolute positive constants depending only on X.
(3) The log log(n) estimate in part (2) is sharp.
P r o o f OF T h eorem 3.1.14. Part (1) is easy: any orthogonal pair will change
K to K\ so that there is an ellipsoid, say £, so that:
(3.1.16) 2“*£ C K x C 2-*£. 33
Then we change £ to a multiple of the Euclidean ball using Proposition 3.1.4.
Part (2): If F is a subspace of Rn with dim F = Xn, by Lemma 2.1.17 (left side inequality),
(3.1.17) v.rad.(K D F) < c(X)a~*~.
Now symmetrize K with respect to F . Since A > | we can take H subspace of
Rn with dim# = Xn and H 1- F. Symmetrizing with respect to H as well, we get a body K\ satisfying:
(3.1.18) K i 0 H C c(X)a^rBXn and
(3.1.19) Kx n # x C c(X)a^r B a _x)n.
(where all c(A)’s may denote different constants depending on A. In fact a closer analysis of c(A)’s show that for | < A < l,they are bounded by a universal constant independent of A, say Co- We do not persist on this though, because later in the proof the constants involved will become dependent on A).
From the last inclusions we conclude that
(3.1.20) Kx C V 2 c{X)a^Bn. 34
Similarly, using the right side inequality of Lemma 2.1.17, the same symmetriza tions give :
(3.1.21) y (# T )'1J,Cifl.
Repeating the procedure (and choosing any orthogonal pairs of symmetriza tions) after m steps we receive a body K m satisfying:
(3.1.22) ( c ( A ) ^ 6(i^A)m)”1Rn C K m C c( ^ a (1r )mBn.
Taking
(3.1.23) m = c0(A) loglog( 7i), will prove the result as max(a, b) < y/n.
Part (3): For this part we need to find a body K and specific sequence of subspaces so that if we symmetrize with respect to these subspaces we will need at least log log n symmetrizations to get close (up to a constant) to a Euclidean ball.
To this end we consider the ellipsoid £ with semi-axes of lengths ai, ( 12,..., an so that
(3.1.24) a 1 — a. 2 — • • • — a a n — ^ Then
n (3.1.26) v.rad.(£) = P J % = 1- j =i Symmetrizing S m times always with respect to the orthogonal pair with first subspace spanned by the last An semi-axes’ unit vectors and second subspace spanned by the first An semi-axes’ unit vectors, we receive an ellipsoid having An semi-axes of length )2"’ and (1 — A)n semi-axes of length —rrr^rrr- rt( A > Thus we will need to repeat the procedure at least m = loglog(n) times in order to bring the ellipsoid to a constant distance (depending only on A) from the
Euclidean ball. □
“Fast” symmetrizations.
We prove now two theorems concerning Schwarz symmetrizations and (1 - A)n symmetrizations with “small” A. By Theorem 3.1.14 we can assume that K has been already symmetrized so that there is constant cq independent of n so that
(3.1.27) Co 15 n C K C c0 B n
Co may depend on A and becomes unbounded only if A tends to 1/2. We have the following: 36
T h e o r e m 3.1.28. Let K be convex symmetric body in Rn with v.rad.(K) = 1
satisfying
(3.1.29) Cq lB n C K C c 0 B n for some absolute constant cq independent of n (cq may depend on X). Then we
have the following:
(1) For all 0 < e < 1 /3 and 0 < A < e, An > 1 there exist less than cn2^ log ^
orthogonal pairs of (1 — X)n-symmetrizations that will change (1.6.1) to:
c ^ B n CKC(l + 12 e)Bn
where c is an absolute positive constant.
(2) For all 0 < e < 1 /3 there exists universal constant 0 < c < 1 so that if n
is big enough to satisfy nxfn < 1 + ce and 0 < A < ce, Xn > 1, there are
3 orthogonal pairs of (1 — A)n-symmetrizations that will change K to K\
satisfying:
{l-e)BnCKl C(l + e)Bn
Remark 3.1.30. The restriction imposed on e, nam ely 0 < e < 1/3, has been added here just to separate A from \ so that we can guarantee that the constant
Co in the relation (3.1.27) can be considered independent of A. In fact instead of
i any other number less than ^ would suffice. 37
We will also prove the following result about Schwarz symmetrizations:
T h eorem 3.1.31. Fix n € N, let 0 < e < 1 be given and let K be a convex symmetric body in Rn , with v.rad.(K) = 1, satisfying
(3.1.32) c~l B n C K C cBn for some absolute constant c. In the following C will denote a universal positive constant that may be different every time it appears.
(1) U p p e r b ound: there exist less than C'^^-log^ orthogonal pairs of
Schwarz symmetrizations that will change K to K\ satisfying:
c ^Bn C Kx C (1 + e)Bn.
(2) Low bound: Let 6 = n"-* — 1, i.e. 9 ~
(i) If the given e satisfies e > there are at most C log ^ orthogonal
pairs of Schwarz symmetrizations that will change K \ to K% satisfying:
(1 - e)Bn C C (1 + e)Bn.
(ii) If the given e satisfies e < , there are at most C ™ log ^ orthogonal
pairs of Schwarz symmetrizations that will change K\ to satisfying:
(1 - e)Bn a 2 C ( l + e)Bn 38
R e m a r k 3.1.33. Theorem 3.1.31 is so far the only result we know that es timates the number of symmetrizations needed to bring a body K close to the
Euclidean ball as a function of the two independent variables e, n (for example, it does not require n to be big compared to a function of e), with both upper and lower e-estimate being achieved after relatively few number of symmetrizations.
R em ark 3.1.34. In fact, in order to achieve the low bound (1 — e) we will need to get first a much better upper bound than the above written (1 -(- e).
The proof of part (1) in Theorems 3.1.28 and 3.1.31 is almost the same and we will present their proofs simultaneously. The proof is based on an iteration argument and the following consequence of Busemann’s formula (see [19] ):
F a c t 3.1.35. For every convex symmetric body K in R” with v.rad.(K) = 1 and for every 0 < k < n integer, there exists a k —dimensional subspace F ofRn satisfying
(3.1.36) vol(K n F) < vol(Bk).
This proposition is used for proving the following Lemma:
L em m a 3.1.37. Suppose K C cBn, 1 < c < cq, with cq being a universal constant, and v.rad.(K) = 1. Then there exist an orthogonal pair of (n — k)- symmetrizations, so that when applied to K will produce a body K satisfying:
(3.1.38) K C c7/2£ ri, 39 where 7 is any number in [0 , 2) such that
(3.1.39) 1^1 > ( ! _ £ £ ( ! _ I ) ) " with k satisfying
<3-L40> where Co is a universal positive constant (and may be a different one every time it appears).
In the case of Schwarz symmetrizations instead of (3.1.39) we use
(3.1.41) >(!__£=(,_!)*)*
Let us postpone for a moment the proof of this Lemma and finish the proof of part(l) of Theorem 3.1.28 and 3.1.31.
P ro o f of p a r t (1), T heo rem s 3.1.28 a n d 3.1.31.
If c > 1 + e, (3.1.40)’ is satisfied if
(3.1.42) k- < 1 £
and (3.1.39) is satisfied if 40 which is satisfied if
(3.1.44) for some absolute constant C\
So, as long as c > 1 + e we can iterate the procedure of symmetrizations described in Lemma 3.1.37 m-times, until
(3.1.45) which is valid for
(3.1.46)
Hence less than cofylogj symmetrizations will suffice. Similarly we prove the part(l) of Theorem 3.1.31 □
We proceed now with the proof of Lemma 3.1.37
Proof of Lemma 3.1.37. Choose F\ Kn with dimFi = n - k so that vol (K fl F\) < vol (Bn-k) (by the Fact 3.1.35). Symmetrize K with respect to F\
and receive a body K\. Take now any F2 F\ with dimF2 = k (this is possible
by (3.1.40)) and symmetrize K\ with respect to to receive a body K 2 such that
(3.1.47) K 2 C c*/ 2Bn. 41
We estimate now 7 :
If x E F2 and |:c| = 1, for t E [0, 1], we have that
vol ((tx + F2 ) H Ki) < vol(\/c 2 - t2Bn-k)
/.\ Z c a — t a
— I vol (y/c2 — t 2 — s 2 Bk)vo\(sSn~2k~1) ds.
After an obvious change of variables, passing to volume radius and using the fact
that vol(J3n_fc) = /q 1 vol(\/l — z 2 Bk)'vo\(zSn~ 2k~ 1)dz^ we get
(3.1.48) v.rad. ((tx + F^) n Ki) <
t [ l ^ _ z2 ^ zn- 2k-l dz Jy/l^/y/d^W ______< y/ c2 — t2 r 1 k 2 ~n—2k — 1 / (1 £ dz Jo
f 1 (1 - z 2Y z n-2k-x dz Jl/c______< y/ c2 — t2 1 -
f 1 (1 - z2Y zn~2k~l dz Jo where for the last inequality we used the fact that < £, Vt E [0,1]. Now,
the function (l - z 2 ) 2 zn~2k~l has maximum at zQ = ^1 — nir|— ^ • Choosing
5 = (l — £) k so that Zq — 5 > £ (a choice that we will analyze later) and using in (3.1.48) the inequality f ^ c(- ..)dz> ■ - )dz, we get 42
v.rad. ((tx + F^) fl Ki) <
To continue we use a few easy inequalities. First
1 - (zo - 5) 2 (3.1.49) ;" v22 ' > 1. 1 Zr\zo and
n—2k — 1 / c \ n > 1 “ ZO
Moreover by the assumption (3.1.40) we can assume that k < j. We use also the definitions of 5 and zq, to get
(3.1.51) v.rad. ((tx + Ff) n Kx) < s/c2 - t.2 ^1 - ^ 1 - ^ where co is an absolute constant.
Putting
(3.1.52) i?2(i) = £2 + w.rad. ((tx + Ff) n Ki ) 2 and
(3.1.53) i ?2 = n?ax R 2 (t) < 1 + v.rad. ((x + F'£) n K \ ) 2 , 43 it is immediate that
(3.1.54) K 2 C R 0 Bn.
Thus the condition Rq < c7//2 will follow from the condition:
(3.1.55) 1 + v.rad. ((® + F^) C\Kx ) 2 < c7, which will be satisfied if
(3X 56) c7 is valid (by the above estimate on the volume radius of (x + F^-) D ifi). The latter inequality is satisfied if (3.1.39) is.
We finally analyze the choice of k.
We required zq — 6 > ~ which leads to zq > j — (n > 2) and this is satisfied if
(3.157) *< (C-1KC + 3) n (1 + c)2 + (c - l)(c + 3) ’ which is valid under the condition (3.1.40) . □
P roof of part (2) of T heorem 3.1.28.
By Lemma 2.1.17 it is easy to see that there exists universal constant 0 < c < 1 so that if n 1/" < 1 + ce, 0 < X < ce (and consequently 1 — ce < AA, for some 44 absolute constant 0 < c < 1 depending on c), there exists no(e) £ N so that for n > no(e) and with dimF — An we have
(3.1.58) 1 — e < v.rad.(K D F 1) < 1 -f e.
Since A < take Fi <-> F-1, symmetrize with respect to F 1 and then with respect to Fi~. If K is the body we received we must have:
(3.1.59) 1 - e < v.rad.(Kr\ Fj1) < 1 + e,
(3.1.60) 1 -e < v.rad.(KnFi) < 1 + e.
From these relations we conclude that
(3.1.61) ^ T Bn ~
Hence
(3.1.62) K D conv ^ ~ ^ B n U (1 - e)S(i_A)„^ .
By Lemma 2.1.12 we get that K contains an ellipsoid of volume radius
(3.1.63) p = (1 - e) ((1 - A)1_aAa) 1/2 (-)= ).
By the assumptions on A (and choosing smaller constant 0 < c < 1 if necessary), we can assume that
(3.1.64) p > l - e 0e, 45 for some absolute positive constant cq. Changing the ellipsoid to a Euclidean ball with one orthogonal pair of symmetrizations we get
(3.1.65) (1 - cQe)Bn C K.
For the above estimate we have
(3.1.66) K C (1 +e)(B(1-X)n x BXn).
But
(l-A )n x 2
£ (1_A)„ X BXn Q£ = {(*i,a; 2 ,...,®n) : X (X - P~2)x1 + X j =1 j>(l-A)n
V /3 > 1 .
For this ellipsoid v.rad.(£) = (3* ^(1 — • Taking fi to be 1/A and using the assumptions on A (again we may need to decrease one more time the constant 0 < c < 1), we get that v.rad.(£) < 1 + cq£ for some universal constant
Co. Finally we use one more orthogonal pair of symmetrizations to change this ellipsoid to a Euclidean ball. □
2.The lower e-estim ate for Schwarz symmetrizations.
To prove the lower estimate we need two more lemmas. The first is a “sub stitute” of the consequence of Busemann’s formula (see Fact 3.1.35 ) but for the lower side: 46
Lemma 3.2.1. Let 0 < 8 < 1 be given. Let 9 = — 1, i.e. 0 ~ Then if K is a convex symmetric body with v.rad.(K) = 1
VF M- Rn with dimF = n — 1 we have the following two cases:
(1) Let the given 8 satisfy 8 > There exists universal constant cq such
that if
K C ( l + c0 6 2 )Bn
then
v.rad.(K n F) > 1 — 8.
(2) Let the given S satisfy 8 < There exists universal constant Co such
that if
KC ^1 + co ^ j B n
then
v.rad.{KnF) > 1 - 8.
Lemma 3.2.2. For 0 < ff < a < 1 we have:
v.rad. (conv(aBn_i U fiBn)) > (3 ^1 + c-^ ^1 - ri* for some universal constant c > 0 .
We will skip for the moment the proofs of these lemmas and we will proceed with proving part(2) of Theorem 3.1.31. P r o o f o f Theorem 3.1.31, part(2).
Let F be a subspace of Rn with dlmF = n — 1. We symmetrize with respect to F and receive body K\. Let a > 0 be such that Ff]Ki — aBn-i. Recall that
Ki D c~lBn. We want now to guarantee that a > 1 — e, based on information on the upper bound.
Applying to K at most log - orthogonal pairs of Schwarz symmetriza tions, we can assume that K satisfies
(3.2.3) K C (1 + c0£2) Bn.
This is done using the first part of Theorem 3.1.31. C is a universal constant, and
Co is the constant in Lemma 3.2.1 part (1). Lemma 3.2.1 guarantees now that for e > yzq we have a > 1 - e.
0+& -I Applying to K at most C n log ^ orthogonal pairs of Schwarz symmetriza tions, we can assume that K satisfies
(3.2.4)
Again, this is achieved using the first part of Theorem 3.1.31, C is a universal constant, and cq is the constant in Lemma 3.2.1 part (2). Lemma 3.2.1 guarantees 48
Let x G F such that |x| = 1. Label with A (see Figure 1) the point on the boundary of Ff)Ki corresponding to the vector ax. Take F\ = (spanfa:})-1-. Now, for every 0 < t < a we will estimate from below the v.rad. ((tx + Fi) fl K\).
The body depicted in figure 1 (that is, the con v(ot(Bn (IF) I) c~xB n)) is inside
K\. The line AB is tangent to the circle c~lBn D span{rc, FL} at B, and T is the
_2 orthogonal projection of B on span {a;}. Let to = £^~ be the length of the segment
O r (see Figure 1)
Case 1 : 0 <<< fo
il-1 dim
F igure 1. Lower bound of the symmetrized body K.
We have
(3.2.5) (tx + Fi) D F i D conv(o!t(tx + (Fi fl F) n Bn) U fit(tx + Fi fl Bn)), 49
for at = \/a 2 — t2 > y/(l — e) 2 — t2 and j3t = \Jc~2 — t2. We also note that
(3.2.6) 4 P - < t < ^ 2 1 — £ at 1 — £
(the left side inequality follows from the fact that the function f(t) = is de-
— 2 creasing for 0 < t < to = — and from the fact c -1 < a).
By lemma (3.2.2) now, we have:
v.rad. ((tx + F{) fl K\) > \/c ~ 2 — t2^1 + C o ^ ^1 — -— (n — l ) 5
> \/c~ 2 — t2 ^1 + Co(l — £ — c- 1)2n*^ "
Thus if c -1 < 1 - 2e we get
(3.2.7) v.rad. ((tx + F{) fl K\) > y / c ~ 2 — t2 (1 + c q £ 2 t i2) " -1 where c 0 may denote a different constant.
Case 2 : to < t In this case we have (3.2.8) (tx + Fi) fl K\ D con v(at(tx + (Fi fl F) fl Bn ) U 0t(tx + Fi fl Bn)) with (3.2.9) fit — ------\fc ~ 2 — t0 2 = the length of HZ a — to >c-- 1~ e~ t \/( l +e)2 - c-2 50 and at as in Case 1. Note also that Oil « (Indeed, we have (3.2.10) & = ‘°2. at + 1 a -to This expression is decreasing for to < t < a, thus (3.2.11) ^ < ^2- < —, » t0 2 using that Thus (3.2.12) u.rad. ((to + Fi) fl K x) > P t(l + c0 n f ) ~ , for some constant Co- R e m a r k 3.2.13. Prom (3.2.7) and (3.2.12) we conclude that the “improve ment” (meaning the increase of the n — 1 dimensional volume of the affine sub space tx + Fi intersected with K x) is slower in Case 1. Hence for the general case we must continue with the estimate (3.2.7). Put (3.2.14) R 2 (t) —t2 + v.rad . ((tx + Fx) D i^ i)2 and (3.2.15) R 0 = info It is immediate that (3.2.16) R 0 Bn C K2. By (3.2.7) and the above remark we conclude that (3.2.17) R o> R {t0). Thus Ro > c- 7 / 2 if R (to) > c~7/ 2. From the latter we get that e e 2 n 2 (3.2.18) t > 2 - c0- n 1 -j- g2n § Hence we will have the lower e— estimate after m steps, where m is such that (3.2.19) c-(7/2)m > i _ 2e, which is satisfied if (3.2.20) m > co ( ^ + log I □ P roof of L emma 3.2.1. Assume that K C (1 +e)Bn. Let F Kn such that dimF = n — 1 We seek now e as a function of 8 that will guarantee (3.2.21) v.rad.(F H K) - 1 - 8 , 52 or better, meaning v.rad.{F fl K) = 1 — r) for 77 < 6 . Symmetrize K with respect to F. Let K\ denote the body we receive and T the cylinder ((1 - 5)Bn fl F) x R. Then (see Figure 2) (3.2.22) i f i C T n ( l + e)Bn. n-l dim 1+e F igure 2. L ow er bo und of v.rad.(K nF). Let also P = Bn \ T and let Q be the layer between the balls of radii 1 and 1 + e and inside the cone defined by the origin and Bn fl T. Let denote the angle between this cone and the subspace F(see Figure 2). In Figure 2 , Q is the region FAZE. Since symmetrization preserves volume, we must have (3.2.23) vol(P) < 2vol(Q) 53 vol(Q) < vol(spherical sector OEZ) - vol(spherical sector OITA) = — (1 + e)n sin + vol(Bn-i) / \/(l + e)2 - i2" dt J(l+e)sin cosn < ((1 + e)" - 1 ) voltBn-O f l + (1 - 5 ) " - 1 Also, pX vol(sector AHQ) = 2 / \ / l — t2 vol(tSn~1) dt J i - s 52 > 2(n - \)vol{Bn-{)-z n* Returning now to (3.2.23), we receive: <3-2-24> '2( r b r ' s H ) t1^ ) which implies: (3.2.25) i52(1 + <5)”-1 < 6e7t2(l + e)”” 1. So, if e (corresponding to the upper estimate on the body K) is given, then there exists 8 satisfying (3.2.25) which will give us the low bound on the volume radius of any (n — l)-dimensional section F (1 K. Let us look differently on this inequality. 54 We in fact fix the low bound (1 - 6 ) we want to achieve for v.rad.(F fl K). So we need, given 5, to find e which will guarantee the desired low bound on v.rad.(FnK). Then it is enough to find the £ — £q that makes the latter inequality an equality. A smaller e than £o will give again the result we want. Let be such that (3.2.26) (1 + ^ ) n_1 = n 2(l + £o)n-1. Then <5o = 6 + £o + ^ o 5 where 6 = — 1 ~ We consider the following two cases: (1) $ > If eo > by choosing e = ^ 8 we guarantee that v.rad.(F fl K) will be even closer to 1 than we need, i.e. v.rad.(F fl K) > 1 — 5 and we are done. Otherwise, we can assume £o < f 5- Now, since S > ^ze we conclude that S2( 1 + So)n 1 < 6£on2(l + £o)n 1 or equivalently S2 < 6eo- Thus we can choose e = c052, with c 0 being an absolute constant, to achieve v.rad.(F D K) > 1 - 8 . 55 (2) S < —g. Then, of course, 52 < Qen2(l 4- e)n_1. x2 Thus we can choose e = cq^-, where Co is an absolute constant, to achieve v.rad.{F fl K) > 1 — 5. □ P r o o f o f Lemma3.2.2. Reffering again to Figure 1, we have that vol (conv(o:Sn_i U (3Bn)) > vol ((3Bn) + vol(spherical prism AEA) > vol (@Bn) + 2(n - l)vol(Rn_i) [ Jb a > vol ((3Bn) + 2 (n - ljvo^Bn-!) , V* ■- [ ( a - s)sn- 2ds V 1 “ (/Va)2 J p Estimating the last integral with (3.2.27) / V (a - s)s^ds > (a- 2 - ± ^ ) (T'* >P and using the fact that to o oo\ 2vol(Bn_i) f 2 [— (3.2.28) 6 .> \ -V n, vol (Bn) V 7T we get 2 s ' 1/n (3.2.29) v.rad. (conv(aRn_i U (3Bn)) > (3 ^1 + ^1 - —n 2 for some universal constant c. □ CHAPTER IV ON THE CONVOLUTION BODY OF TWO CONVEX BODIES 1.Mixed Convolution Bodies And Examples. Definition 4.1.1. Let K and L be two convex symmetric bodies and 0 < 8 < 1. We define the convolution body of K and L of parameter 8 to be the set (4.1.2) = : vol(KD(x + L))>5vo\(KnL)}. It is obvious that C(<5; K, L) is not affected if K and L are interchanged in the definition. C(8 ; K, L) is symmetric and it is again a consequence of Brunn- Minkowski inequality that this set is convex. It is also immediate from the defi nition that if <5 —>■ 1 then C{8 \ K, L) “shrinks” down to zero provided that for all x€Kn \{ 0}, vol(Kn(x + L)) < vol(ii:nL). Note that the boundaries b(C(8 ;K,L)), 0 < 8 < 1, are the level sets of the standard convolution of the characteristic functions Xk (x) and Xl(x) of the sets K and L. 56 57 We will look for the “right” normalization exponent a, such that the limit: <5->l (1 - By “degenerate set” we mean a set with empty interior. The case of an infi nite cylinder is considered to be a non-degenerate case and it may happen to be even Rn. We understand convergence as convergence (in Hausdorff sense) of the intersections of our sets with any (fixed) Euclidean ball in Rn. There are easy examples where for 0 < ao < 1/ 2, the limiting convolution body collapses to a point for all exponents a < a 0 and it converges to an infinite cylinder or Rn, for all a > ao- In particular theorem 2.2.5 says that: (4.1.3) C{K,K) = P°k . The body C(K,L) when it exists, is also affine invariant since it is easy to see that if T is any invertible linear transformation of Kn then, (4.1.4) C(T{K),T(L))=T{C(K,L)). 58 When it is clear to which bodies K and L we refer to, we abbreviate C{5-,K,L) to C(<5). The following example describes this body in the case that K is the unit ball of and L the unit ball of ££ for 1 < p < oo. E x a m pl e 4.1.5. Let 1 < p < oo, 0 < 5 < 1 and C(S) = {(xi,X2 , .. .,xn) = x E Rn : vol((x + Bgn) fl Bgn ) > Svol(Bgn)} Set n 4~P-— = 1 forp = oo. Then we have: (4.1.6) C (S £3o,£*n) - lim _ A n v B t * ^ , where Anjp is a constant depending on n and p. It is uniformly bounded and separated from zero for n € N and 1 < p < oo, and its exact value is _ ( n + p - l)i*S=T / wKBt*) V - 1-"-' p { vol(Bi r ,)) ,0 r P * °° — 2 for p = oo R e m a r k (4.1.7). It is interesting to note that convolving the unit ball of £ with itself gives (up to homothety) the unit ball of i™, however, although is almost isometric with (Banach-Mazur i) < ~ 1 + ) when we convolve their unit balls the result is (up to homothety) the unit ball of £%. 59 Proof for Example 4.1.5. For p = oo the result follows from the result of M.Schmuckenschlager (see [25]), since it is well known that the projection body of B e is ^ BeSo. Assume 0 < p < oo. Let x — (xi,X2 , ■ ■. ,xn) be a point on the boundary of C(8 ). If 8 is close enough to vol(B*n) and since e* 4- ej fi projsp&n{ei,ej}(Be »), x will satisfy n - i vol((ar + Btn) n B t « J = ^ / vol((l - t ^ B ^ - i ) dt = (1 - 8 )vol(Ben), which implies n f 1 „-i (4.1.6) vol(Ben-i)y^ / (1 - tp)~ dt = (1 — £) vol (I?*!.). By the intermediate value theorem for every t G [1 — |a?3-|,l] there exists & € (1 - |x,-|, 1) so that 1 — tp = p£f-1 ( 1 - t) and the above becomes: f 1 n— 1 1 n -1 (4.1.7) v o l ^ - i ) ^ / p— ^ p (1 - 1)— dt = (l-5)vol(B,;;). J = 1 I ® J I Using the information on , Bernoulli’s inequality and finally evaluating the integrals we get: 2L n+p—1 ™ . . n + i >-1 P _ (.-pfr-i) max . a < "+p-i v°1(^ > < 5 ^ p l Observing that letting 5 —> 1 all Xj converge to zero we get our conclusion. □ 60 We also give the following example: E x a m p le 4.1.8. The convolution body C( 6 ) of Bq and £Bsatisfies: (4.1.9) where n\ 1/r < 1} and 21/n (n!)1/” constant. n Proof. Let x = (®i,a;2, • • • ,a;n) be a point on the boundary of C( £j = ± 1, Vj = 1, 2,...,n. Each 1—dimensional side of each such simplex (the ones parallel to the axis) has length equal to: j=i 61 Consequently, vol (B£?\(x +B eno)) = ^ Y , ^ f l £rx 3 j (4.1.10) from which we get: n E E £ j x j 6j=±l j=i 2n+ 1n! (4.1.11) 1 -S TV The result now follows. □ We continue with introducing some terminology first: Definition 4.1.12. Let K and L be two convex symmetric bodies in Rn. We say that the body L is well fitted in the body K if the following two conditions hold: (1) L C K (2) \/x &W1 , x 0, we have x + L K Observe that the number of touching points of the boundaries of K and L in the above definition can be even two. This is the case for example with the bodies Ben and the ellipsoid with one semi-axis of length equal to 1 and all other semi-axes of length equal to 2. 62 We have the following: PROPOSITION 4.1.13. Let K be a (convex symmetric) polytope such that Bgn is well fitted in K. Then the limiting convolution body of K and the Euclidean ball C(K, Bin) is a section of Bg™ for some m e N. " 2 ” P r o o f . Let ±z\, ±Z2,..., ±zm (for some m e N ) be the touching points of K and Bgn. A similar computation as the above gives that if a: is a point on the boundary of the convolution body C(5) it satisfies m \ ^ X | < x, Zj > | "2 I 2 / \ ~ir (4.1.14) lim 1 * -(» + !)-" f «*(*;)'- —► 1 ^ ___ 8) n + 1 ^ \fVOl(Bgn-l) Thus (4.1.15) lim — — ^ 2"• = °nL, <5->i (1 _ ,5) n + l where 2 n + l ,n ± l ’ (4.1.16) L = { x e R n : | <*,*,•> I'’ J < 1}. □ T h e o r e m 4.1.17. Let K be any convex symmetric body in which Bgn is well fitted. Then for every e > 0 there exists a convex symmetric body P in which Bgn 63 is well fitted, so that d(K , P) < 1 +e and C(P , Bgn) is almost isometric (up to n* ) to a multiple of an affine position of Proof. It is enough to show that the body P can be chosen so that it has only 2n touching points ±zi, ±Z2 ,..., ± 2n with the Euclidean ball, with z\, Z2 ,..., zn being linearly independent, and in a neighborhood of each zj,j = 1,2,..., n the boundary of P is flat. Then the result will follow from an analogous computation like in proposition (4.1.13). Let k > 1 be the maximal number of linearly independent vectors on the common boundary of K and Bgn. Let ±zi,±Z 2 ,..., ±z/c be such a set. For each x on the boundary of K let x* be the supporting functional of K at x. Let 9 > 0 and eo > 0 such that (1 + eo )2 < 1 + e and (1 + ^)n < 1 + £o- Choose Zk+1 6 Sn_1 D (span{z 2) 23,..., z*})'1' so that Zk+ 1 is linearly independent of z\ and so that if K\ = (K D {a: € Kn : |zjj+1(x)| < 1}) we have: (4.1.19) d(K, Ki) < 1 + n This choice is possible since as Zk+i gets arbitrarily close to z\, K\ gets ar bitrarily close to K. We continue this procedure (choosing next Zk+ 2 £ § n-1 (~l (span{z 2 , 23,..., Zfc+i})"1 so that Zk+ 2 is linearly independent of z\ and if K 2 = (Ki D {a; € Rn : |z^+2(a:)| < 1}) then d(Ku K2) < (l + f)), until we end up with a body K = Kn-k that its common boundary with Bg» has n linearly 64 independent vectors zi, Z2 , •.., zn, and (4.1.19) d(K,K)< (l + £) < 1 + £q. Now let s > 0 and set (4.1.20) K a = |^(1 + s)K n p.{a; 6 Rn : \zj(x) \ < 1} It is clear from the definition that in the Banach-Mazur distance (4.1.21) lim KS = K. s —>0 Choose so small enough so that d A”So j < 1 -f £o, and put P = KSo. □ 2.The Main Results. We start by introducing some additional notation. Let A be the Lebesgue measure in Rn or restricted to n — 1 dimensional sets (hopefully it will cause no confusion). For y on the boundary of the convex symmetric body K we denote by N(y) the normal vector of b(K) at y if it exists (and it is unique). By convexity the normal vector exists for a subset of b(K) of full measure (see[26]). For a function o> ■> where X{...} denotes the characteristic function of {••.}• 65 We will try now to identify the limiting convolution body of two convex bodies under some certain conditions. Before we continue with the statement of the theorem, let us note that the norm of the polar of projection body of the body K, is given by the following: (4 .2.1) VueS”- 1, ||u ||^ = -± — 1 \ The last expression is clearly a norm and it also shows that the projection body is always a zonoid. A modification of the previous expression will appear in the next theorem. The quantity [ < N(y),u>+ d\{y), Jb(K)r\b(L) for u € S” " 1 defines, by homogeneity, a norm (the triangle inequality by (< N(y),ui > + < N{y),u 2 >)+ < < N(y),ui >+ + < N(ij),u 2 >+), provided that b(K) n b(L) is “rich” enough: F a c t 4.2.2. Let K be any convex symmetric body in Rn , and let A be any subset of b(K) , satisfying: (4.2.3) A ({ye A : < N(y), u > > 0}) > 0 Sn~\ Then for uGE", the quantity (4.2.4) |u|voln -1 (proj[u]±{y € A : < N(y),u> > 0}) , 66 where |uj denotes the Euclidean norm of u, defines a norm. P r o o f . □ The above norms can be obtained through the mixed convolution bodies. We have the following: T h e o r e m 4.2.5. Let K , L be two convex symmetric bodies, so that L is well fitted in K. Assume that b(K ) n b(L) has countably many components and (4.2.6) A(y € b(K) n b(L) : < N(y),u > > 0) > 0 Vu € Sn_1. Then the limiting convolution body C(L,K) satisfies: (4.2.7) C(L,K) = lim s—>i_ l — o and its norm on Sn-1 is given by: (4.2.8) \\u\\C (l,k) = vol(L)~1 vol(proj[w]x{y € b(K)C\b(L) : < N(y),u>> 0}), for all u € Sn_1. In particular if K = L then the above result gives that the limiting convolution body of K is the polar of its projection body. P r o o f . Let us assume that OO (4.2.9) 6(if)n4(£)=U(±Vj), J '= 1 67 where ±Uj for j €. N are the components of b(K) fl b(L). Let u € 8 n_1, and for 6 > 0 let ko = ko(9) be such that OO (4.2.10) A |projHJ. I IJ j < 0 . For 7 > 0 set (4.2.11) Ujtl = {x € b(K) : dist(a:, Uj) < 7}, where dist is, say, the geodesic distance on b(K). Clearly (4.2.12) UmA(tlJlT\Cli)=0,- for all j € N. For 0 < t < 1 we construct the following truncation of L: Consider the cylinder T = {Rii}+L (Minkowski addition) and set Lt = (tT)r\L. Lt is convex symmetric body, (4.2.13) proj[u]j.r = proj[u]xL and (4.2.14) lim Lt — L t—► 1 (say, in the Banach-Mazur distance). By convexity and compactness it follows as well that (4.2.15) ^lim voln_i (proj [u]±(b(L) \ Lt)) = 0 68 and (4-2-16) yEb(Lt)\b{L)J ^ u r S len9th^ y + Ru ) n (h(Lt) \ b(L))} > °* Thus if S is close enough to 1 (i.e. 8 (t, 7 ) < (4.2.17) I < (1 - 5)|H |c(ill.l1K)vol(L1) < 1 + 11, where (4.2.18) 1= f < N (y ),u > + dX(y) Jb(Lt)nb(K) and (4.2.19) 11 = [ The last inequalities can be rewritten as (4.2.20) I < \\u\\as\L,,K) vol(Lt) < I + II- Taking now 5 —>• 1“ the liminf and the limsup of the norm are bounded between I and II. Let 7 go to zero and then 9 go to zero. Then the integral II will converge to zero (since the domain of integration converges measure-wise to zero and the 69 indegrant is a bounded function ). This proves that the limit of the latter norm exists and equals the integral I . Thus we have arrived to : (4.2.21) IM|c(Lt,iC) = } ( r , [ < N{y),u>+ d\(y). vol(Lt) Jb(Lt)r\b(K) We want now to let t —> l - . The right hand-side of (4.2.21) converges to : (4.2.22) Zntfn f (4.2.23) voi(L )pr°^ u]± ^ G b^ : < N ^ ’U > ~ °^* It remains to show the following claim: Claim. limt_*i- \\u\\c{Lt,K) = \\u \\c (l ,k ) [Allowing volume (1 — 5)vo\(L) to go out of K by shifting L in the direction of u, say that the shift is achieved by the vector agu and volume (1 — <5)vol(Lt) goes out of AT by a shift of Lt by the vector fyu. Define for convenience the following quantities: Ib{Lt)nb(K) vol (proj[u]L(b(L)\Lt)) (4.2.25) A = 1 + Ib(Lt)nb{K) < N{y),u >+ d\(y) If L is shifted by the vector ({3$ + (1 — 5)T)u the volume that gets out of K is at least (4.2.26) (1 - S)vo\(Lt) + (1 - S)T f < N(y), u >+ d\(y) Jb(L,)nb(K) and this equals (1 — 5)vol(L). Hence we conclude that (4.2.27) <*«<& + ( 1 ~ S)T, provided that 6 is close enough to one. Similarly, if Lt is shifted by (a^A - (1 - 5)T)u the volume that gets out of K is at least (1 - <5)vol(L) - ajvol(proj [u]±(b(L) \ Lt )) + (a6(A - 1) - (1 - 6 )T) I < N(y),u > + dX(y), Jb(Lt)nb{K) which equals (1 - 8 )vol(Lt). Consequently (4.2.28) fa < a5& - (1 - <5)I\ provided that 5 is close enough to one. Combining (4.2.27) and (4.2.28) we get (4.2.29 ) ft + (1 ~ f )F < a, < ft + (1 - (4.2.30) —— = (1 - 5)\\u\\C(6.LiK) = | M | ip c*<5 i-« and similarly for /3$ and letting S —> 1~ in the relation (4.2.29) we get that the quantities lim inf^x- l|u|| c(s-,l.k) and lim su p ^ x - Hull c(s-,t io are bounded 1—5 1—5 between the quantities A 1_____ l r and 1 1_____ uf' IMIc(Lf,K) Since this is true for all 0 < t < 1, letting t -» 1~, and noting that T goes to zero and A to one, we get that the limit of I lull c(s-.l,k) as d —>■ 1“ exists and it is l-S equal with lim ^x- |Mlc(Lt,j?) proving the claim, and the existence of the mixed limiting convolution body of L and K as well. □ R e m a r k 4.2.31. If b(K)nb(L) is strictly convex, then the condition (4.2.7) is automatically guaranteed if we just assume that the Lebesgue measure of b(KnL) is positive. Of course this is also the case when K or L is strictly convex. R e m a r k 4.2.32. In the case that K and L are not symmetric, theorem (4.2.6) is valid although the relation (4.2.9) does not define a norm but the Minkowski functional of C(L,K). C oro llary 4.2.33. Let K = B n be the Euclidean ball in Rn of radius one. Let A be a closed subset o fS n~l, symmetric with respect to the origin and such that A (A) > 0. Then, for every invertible linear operator T : W 1 —tW 1 we have: vol (proj[Tu]± (T ({x e A : (N t K(T x ),T u) > 0}))) I * * (7 ) I vol (proj[u]± {xeA : (*,u> > 0}) where u € Rn and | . | denotes the Euclidean norm. P r o o f . Let L be the convex hull of A. Then since for every invertible T : W 1 —» Rn we have (4.2.34) T(C(K,L)) = C(TK,TL), we conclude that (4.2.35) \\Tu\\C(TKtTL) = \\u\\c(K,L)- Now we get the result by using the special form of the norm of C(K,L) and C(TK,TL) described in theorem 4.2.6 □ We continue with the theorem for the case that the two bodies K and L have a n — 2 dimensional manifold as the intersection of their boundaries. This, in a sense, may be considered to be the “general case”. 73 T h e o r e m 4.2.36. (Ellipsoid Limit Convolution Theorem). Let K and L be two symmetric convex bodies that satisfy the following conditions: (1) vol((x 4- K) D L) < vol(K fl L) for all x E R" \ {0}. (2) The set M = b(K) 0 b(L) is an (n — 2 )-dimensional manifold such that the outer unit normal vectors N k(x) and Nl(x) of K and L, respectively, exist (and they are unique) for all x € M , and their restriction on M is a continuous function ofx. Moreover, if X is the (n — 2)-dimensional volume measure restricted on M , then 0 < X(M) < oo. (3) M contains no “touching points”, i.e., there is no x € M for which NK (x) = NL(x). Then the limiting convolution body of K and L is an ellipsoid (possibly “degen erated”; see remarks 4.2.39 and 4.2.40^. In particular (4.2.37) CiK,L)=lnn_f^§ and Hull ( 1 f (u>NK^ ^ NL^d\(x)]l/2 (4.2.38) IMI C(K,L) - { 2 v o i { T n T ) j M \NK(x) x NL(x)\ dX{x)) ’ for all u e R n, where \Nk {x ) x N l (x )\ denotes the length of the cross product of the two normal vectors N k (x ) and N l (x ) of K and L respectively, at the point x, that is, the quantity (l — (Nk (x ), Nl (x ))2) . 74 The conditions required by the theorem are not really restrictive as they are satisfied for the “generic” bodies K and L. For an example consider K = pBi* and L = Bin for 1 < p < <7 < oo, 1 < p < tip- * and p ^ kp~q, k = 1, 2, . . . , n. For bodies whose boundaries intersect on an (n - 2)-dimensional manifold but not neccesarily C1, can be included in some cases in the Ellipsoid Limit Convolu tion Theorem through special approximation arguments. In particular, the result of the theorem is true for polytopes. R e m a r k 4.2.39. It may happen that |j . j|c(K,L) is a seminorm, which means that the ellipsoid “degenerates” to an infinite cylinder. There are examples in which the limiting convolution body degenerates to Rn for all exponents a > 1/2 and collapses to a point for all a < 1/2 (see Example 4.3.64). R e m a r k 4.2.40. For “generic” bodies (4.2.38) defines a norm and not a semi norm, which means that the limiting convolution body is a non-degenerated ellip soid. However, in the case that (4.2.38) defines the zero seminorm, the behavior of the convolution bodies is not so regular : there are examples for which the limiting convolution body, for suitable normalization, can be very different, for example, a cube (see Example 4.2.72). In the case that (4.2.38) defines a non-zero seminorm (but not a norm), then it is not possible to receive a non degenerated limiting convolution body without using different normalizations in different directions. 75 R e m a r k 4.2.41. If the convolution / of the characteristic functions xk and X l of K and L respectively, is of class C 2 at the point zero, then the indicatrix of Dupin exists for the graph of the function / at the point zero, it is an ellipsoid (see for example [26] ), and this ellipsoid is the limiting convolution body of K and L computed in the above theorem. However, the level of smoothness of / at zero is not known a priori. From the final stages of the proof of the above theorem, one can see that / is C 1 at zero. In the case K — L the function x k * X l behaves as a cone at zero and so it is not even differentiable. However, in this case the limiting convolution body can be an ellipsoid. Before we present the proof of the theorem, let us introduce some additional notation. We write Nk (x) and Nl(x) for the outer unit normal vectors of b(K) and b(L) at x respectively. For an arc AB we write AB A triangle defined by the points A, B,F is denoted by ABT^] the angle at B is denoted by ABT, at V -*s. —* —* with ATB and so on. Finnaly we write \AB\ for the length of the vector AB. Let us also recall that a tubular neighborhood V of a fc-dimensional manifold M in R” for which a normal n - /^-dimensional subspace Fx exists at every point x € M, is an open neighborhood of M such that for all x,y G M with x ^ y, (Fx Pi V) and ( Fy fl V) are disjoint. The existence of a tubular neighborhood is guaranteed if M is a C 1 compact manifold (see for example [10]). 76 Proof of Theorem 4.2.36. Let u e R n and let Uj, j £ N, be the connected components of the set {x£b{K )f\L : (u,Nk {x)) > 0}. By shifting K by the vector u0 = ||u||c“ k € b (C(S; K,L )), K “captures” some volume from L in the directions Nk(x) for x £ Uj, j £ N. Simultaneously some volume from L that was inside K is lost and this is in a neighborhood of the -Uj's in the direction Nfc(x) for x £ -Uj, j £ N. Since by symmetry (4.2.42) -[(K n L )\ ((K - x ) n L)) = ( K n L ) \ {{K + x)nL), the loss of volume from L that occurs when we shift K by uq is the same with the loss of volume when we shift K by —uq that occurs in a neighborhood of Uj in the directions —Nk {x), x £ Uj. Thus in order to compute the loss of volume (4.2.43) (1 — 8 )vo\(K D L) = vol(Jir C\L) — vol((uo + K)C\L) we have only to look at a neighborhood of the Uj’s in the directions Nk(x) and —Nk(x) for all x £ Uj for all j £ N. Let 9 be a positive number and Ujtg be a ^-neighborhood of Uj on b(K), i.e., (4.2.44) Ujtg = {x £ b(K) : geodesic-distanc e(x,Uj)<9}. We may consider that we have only a finite number of Uj's by considering only the first m of them with m £ N such that A (u JLm+iHUj)) is less than a given 77 number, say e, and use an approximation argument letting e go to zero. Then 6 can be chosen small enough so that the Uj^'s are all disjoint. For 5 close enough to 1 we may write: (1 - (^vol^ n L) = ^ 2 (vol {K DL)~ vol((u 0 + K) n L)) j = 1 2 f vo1 f U W if i - *“°) n Ir) I - vol [ U + tu0) n L) j \ \0 = ^ 2 ( vo1 ( U W i f i ~ tu°) n L) ) - vo1 ( U W j , e + tu°) n L ) ~ u o j \ \0 (4 .2 .4 5 ) Titj = [(UjtQ - tuQ) !1 L]\[((Uj,e + tuQ) (1 L) - uQ] 0 (4.2.46) T2j - [((£/?,e + tu0) D L) - u0] \ [(Uj,e ~ tu0) H L]. 0 Fix j € N and label with A a point x in the (n — 2)-dimensional manifold M ij = b(Ujte fl L). Let 5,T, A be the intersection points of the 2-dimensional plane spanned by the vectors Nk (x ) and N l {x ) with the n — 2-dimensional manifolds M 2j = b ([(Ujtg + «o) H i] - u0), M sj = b ((Ujtg fl L) — uq) and 78 M4,j = b {{Ujfi — uq) fl L) respectively, so that the plane figure defined by the points A,B,Y,A and the boundaries of K, K — u q ,L ,L — uq on the Sx = span ( N k (x ), N l {x )) is inside T ij or T2 ij. Being in T ij or T2j depends on whether (N k (x ), u o )(N l (x ), u o ) > 0 or (N k (x ), u q ){N l (x ), u q ) < 0 respectively. In or der to compute the volumes vol(Tij) and v o l^ j) we will integrate on M the volumes of the plane figures ABYA as described above, for all x £ M (all j ’s) (here we make use of the existence of a tubular neighborhood of M in l n). We concentrate now our attention on the two dimensional section x + Sx. Let E be the point of intersection of the line BY with the tangent line of b(L) at A ; Z the point of intersection of the line TA with the tangent line of b(K) at A; H the point of intersection of the arc AB''' with the tangent line of b(K - u q ) at T; and © the point of intersection of the arc A Awith the tangent line of b(L — u q ) at Y. These intersection points exist for <5 close enough to 1, by the compactness of M and the condition (3) of the theorem. Let us write ABY'~' for the region defined by b(L)C\(x + Sx),b(K-uo)C\(x + Sx) and Ar, and by AAr'-' the region defined by b{L - uq) fl (x + Sx), b(K) fl (x + Sx) and A r. Clearly, (4.2.47) AYH* C ABY~ C AYE a (4.2.48) Ar©A c ArA~ c a y za . 79 Thus, vol( ATHa ) + vol(Ar©A) < wo\{ABTA) < vol(ATEA) + vo\(ATZa ). Now, (4.2.49) vol(^riBA) = ilAJSIKiT.AWA))!, (4.2.50) vol(^rZA) = ^AZHiAT.NiciAM, (4.2.51) vol(ArffA) = i|Aff||(/fr,JW 4))|, (4.2.52) vol(>lreA) = i|i4&||(AT,JVAe(il))|, where for a segment AB we write, say, Nab(B) for the unit normal vector of AB at B with direction such that has a positive inner product with x. Note that as <5 —► 1" the normal vectors converge to either Nk(x) or Nl(x) (Nah(A) converges to Nl(x), and Naq{A) converges to Nk{x)). Using now the law of sines to estimate \AE\, \AZ\, \AH\ and |-A0|, we get: 1 / 1 (AT, Nk-uq (T)) { ir , Nah (^)) I 2 ^ |N k . U0 (T) x NAh(A)\ 1 {AT, Nao(A))(AT, NL- Uo(T))| \ \Na@(A) x Nl- U0(T)\ ) < vol(ABTA) < 80 1 ( \{AT,NBr(T))(AT,NL(A))\ 2 \ |^r(r) x N l (A)| | (AT, N rA (17)) (AT, N k (A)) | \ |ATrA(r) x Nk (A)\ j • Observe that r € M — Uq and tends to x as S tends to 1. In order to finish the proof we need the following claim: C laim . The vector ||u||(7{<$;tf1L).Ar converges (say, in the Euclidean norm) to Px(u), where Px is the orthogonal projection of Rn to Sx. Let now j) : M —» {±1} be defined by (f)(x) = 1 if ABTA C T \^ ,j = 1,2,..., and |u "22c(s,k,l) vo\(K C\L) — - f IIwllc7(<5-,k l)V ~ n I \\u \\c(8;K,L)V0K A B T A )d x + |Mlc(,5;K,L)0(77“p )> 2 7^-i({_l}) INI c(6tK,L) where no ambiguity arises if we recall that ABTA depends on x. We now use the estimates on vol(^4SrA) and take the lim sup^! and liminf^_n in the above equation. Using the monotone convergence theorem, and the Fatou’s lemma, 81 we get our result since (Px(u),NK(x)) = (u,NK(x)) and (Px(u),NL(x)) = (u,Nl(x)). Proof of Claim: Let P be the projection point of x + u 0 on x + Sx and U be the terminal point of the vector x + u0. Then AP = x + Px(u0), |AT - AP\ = |PT|, thus we must prove that ||u||c-( (4.2.53) |rT/| = sin(r 1------. sin(ArU) Consequently, (4.2.54) ll«llc(Si«,t)|Pr| = coS(ffi7). sm(ArU) Observing that by the condition (3) of the theorem ATU is tending (as 5 -> 1) to ^ and PTU converges to ^ we finish the proof of the claim. □ We continue now with the case of poly topes. For x on the boundary of a convex body K we write N (K ,x) for the normal cone of K at x (see [9]). T h e o r e m 4.2.55. Let P, Q be two convex symmetric polytopes satisfying: (1) voln (P n (i + Q)) < volniPnQ) for all x G l \ {0}, (2) P and Q have no “touching points”, i.e., if x G M = bd(P) D bd(Q) then neither N (P, x) C N(Q,x) nor N(Q,x) C N(P,x). 82 Then the limiting convolution body of P and Q exists and it is an ellipsoid. For a convex symmetric body K we will write extrK for its r-skeleton and for a set A C R, x* G R and e > 0 we will write Wa (x *',s) for the set {x 6 A : |(a:*,®)| > 1 - e } . Sketch of proof of Theorem 3.13. Let Z = (M n extn_2P)u(M n extn_2Q). Clearly M \ Z is a C 1 (n — 2)-dimensional manifold. Let Fi, F2,..., F/v be all the (n — 2)-dimensional faces of P and Gi,G2, ... ,G r all the (n - 2)-dimensional faces of Q for som e N, R G N. For each F{ and Gj (1 < i < N, 1 < j < R) let x* and yj be vectors in bd(P° ) and bd(Q° ) respectively so that each x* belongs to the interior of the normal cone of Fj, each y* belongs to the interior of the normal cone of Gj, and for all 1 < i < N, 1 < j < R the vectors x* and yj are not collinear. Consequently, (1) extn_2P C WP(x*j-,e) and ext„_2Q C |J ^=1 WQ{p*j-,e), (2) An_i (UyLi Wbd(P){Xj',ej) < ce and pn-\ (u jL i w bd(Q){yj',e)) < where \ n-i and pn- i denote the Lebesgue measure of K restricted on bd(P) and bd(Q) respectively. We construct now convex symmetric bodies Pi, P2,..., P/v, Pe and Qi, Q 2 , • •. ,Qn,Qe by successively truncating P and Q (Pi = Pi(e) and Qj(e)) in the following way: set first Pq — P, Qo = Q and assume that the bodies P j-i and Q j-i have been defined. For each vector Xj we examine whether 83 eventually (for e —► 0) the set Wp(xj] e) fl ext n_2Q is empty or not (it can not be frequently non empty). If the above set is eventually empty then we set Pj = Pj-i \ WP{x*-,e) and Qj = Qj-i- If not then let {y*^, y*2,..., y*jm .} be a subset of {2/1 > 2/2 ?'•*■> U r} of minimal cardinality ( rrij G N) so that rrij W p(xj; e) n ext n_ 2<3 C 1=1 for all e > 0. Then we define Pj = Pj-i \ Wp(Xj-,£) and rrij i— 1 where 0j,i(e) > 0 are chosen so that the sets {a; G P j - i : |(x|,a:)| = 1 — e} and {V ^ Q j-i '■ I(Vj,ii v)\ = 1— intersect on an (n — 2)-dimensional manifold for all £ > 0 (here we use the pairwise non collinearity of the x*'s and y ^ s). We continue inductively until all a;J, x \,..., x*N are used. Finally we construct P£ and Qe. Set Pe = PN. Let {yjv+ lil, • • •, vn+ i.mw+i} be the subset of {y{, ...,y*R} that consists of the vectors that have not been used in the above inductive procedure (it may be empty). Set ms+i Qe^QnX WQ(yfljj_imN+1;e). i=i We can now use the same arguments used in the proof of Theorem 4.2.36 in order to prove the following inequalities; we write Me for bd(P £) fl bd(Qe) and 84 C(£) for C(S; P, Q): \ f j|'u||^(5)vol 2(i4BrA)dx - l [ \n\c{6)v°h{ABTA)dx < M l2 CIS) V0ln-2(-PHQ) < (i-s) 1' 2 \ [ llullc(5)Vol2(^4-SrA)dx ~ l [ M\c(5)v°h(ABrA)dx + i I Ilullc7( + \ f ll“ llc(6)vol2 (UjT 2,j n (x + Sx)) dx , where 5"x is the 2-dimensional subspace of E orthogonal to M2e at x. Clearly, by the same arguments in the proof of Theorem 4.2.36 and the construction of M2e the last two integrals are less than ce for some constant c > 0 independent of e and 5. Passing to the limits lim supj^ and lim in f^i and then lime_ 40> the above inequalities give the result. □ We observe here that the result in 4.2.36 is much more “unstable” than that of 4.2.6 in the following sense: 85 T h eo r em -E x a m pl e 4.2.56. For every 1 < p < 2 for every n > c (c is a universal constant) and every 0 < e < \ there exist convex symmetric bodies K, L , L and points ± ei, ± e 2, • • •, ±en on the boundary of K, so that (1) b(K) n b(L) = (b(K) n b(L)) U {±elf ±e2, ...,±enj (2) LCLC(l + e)L (3) C(K , L) is an ellipsoid (4) C(K,L) is (up to homothety) Bin. P r o o f . Let 1 < r < \/2 and consider the sets Lq = rB q and Kq — B i^. Assume that b(Lo) is equipped with the normalized Haar measure p. Let A = rS n_1 \ Kq. It is well known that there exist constants Ci,C2 so that fi(A) < Cie~c'jn. Thus by a standard concentration of measure argument we get that there exists universal constant c, so that for n > c there exists an orthonormal basis ei,e2,. • •, en of Rn, satisfying rej A for all j = 1,2, ...,rt. Let a = min{l, dist(rej, conv(A)), j = 1, 2,..., n} > 0 and for 0 (4.2.57) L(t) = rB q \ |J{a; € rB q : |(x-, ej)| > r( 1 - te)}. i Let L = L(a) and (4.2.58) K = B i^ \ (J{ x € Be^ : [(x, ej)\ > r(l - y)}. j It is clear that (4.2.59) b(K) n b{L) = b{rBtn) n b(Be,J , 86 so that C(K,L) is an ellipsoid (actually it can be shown by direct computation that it is a Euclidean ball). Let (4.2.60) L = conv (L (a) U r|l- ’ where q = an(l here the ball Bi* is understood to be expressed with respect to the basis ei, e 2,..., en. Then according to the theorem 4.2.36 and example 4.1.5 for S close enough to 1 and u £ § n_1 we have that 1 f (u,NK (x))(u,NL(x)) 2HtiHc( + Cp,»X)l(irir^ - (! ~ 8 )vo\(K n Z), j llull c {S\k X) where Cv NuHc(g;y£) f (u,Nk (x))(u,Nl(x)) 2WUWc(6;K,Z) Jb(K)nb(L) INk(x) X Nl (x)I + ~ volfA"n n (1-«)1/P Since p < 2, if we let 5 -» 1 , the fraction of norms in the first side of the above equation converges to zero, thus we get that 87 (4.2.61) and (4.2.62) C(K,L) = We conclude by observing that (4.2.63) U C ( 1 | £)L, 1 — ae if e > 0 is chosen to satisfy 0 < e < \ . □ We now present two examples describing the situations discussed in remarks 4.2.39 and 4.2.40. Our examples are given in R2, but revolution of the described bodies with respect to the ei-axis and e 2-axis in Rn , will produce similar examples in higher dimensions. E x a m pl e 4.2.64. For any 0 < ao < 1/2 we construct bodies K and L in R2 so that, and 88 as S —> 1 . C onstruction . For all 0 < ag < 1/2 consider the function, = / l i (xl/a° (loS * ) ’ for a; > 0 f(t) I 0, for x < 0 . Consider now K to be the body bounded by the curves: 2/ = 1 y = - 1 y = 1 - f(x — 1), for x > 1 y = - 1 + f(x - 1), for a; > 1 y = 1 — / ( —x + 1), for x < 1 k y = - 1 + f ( - x + 1), for x < 1. Let R denote the rotation of R 2 by 7r/2 (say with det(R) = 1), and let L = R{K). We distinguish now two cases: Case 1 a > ao- Observe that by symmetry it is enough to show that the norm of the (standard) basis vector e 2 with respect to the body C{5; K, L)/(l — 6 )a converges to zero as 5 —> 1 ~. Denoting for simplicity the set C(5',K,L) with C(S), we have: (4.2.65) (1 — <5)vol(-fiTDL) = 2 f C(S) f(x)dx , Jo hence, -1 (4.2.66) M l l , c w = d M I C (6 )° (loglle 2llc(<5)) (T-Tr5" " Since 1/a — 1/ao < 0, we get the result. □ Case 2 0 < a < ao* Fix an angle 0 < 9 < 7r/2, and let (xi,X 2) be a point on the boundary of C(8 ]K,L ) so that x \/x 2 = tan#. Assume without loss of generality that X\,X 2 > 0 (the cases x\ = 0 or X2 = 0 are easy). Then for r = y/x\ + x\ it is enough to prove that uniformly as <5 —> 1 . Shifting K by ( Xi,X2 ), we easily get that, fx2 fx2~f(xl) (4.2.68) / f(t)dt + / f(t)dt < (1 - (i)vol(A: n L). Jo Jo By computing the integrals passing to polar coordinates and using the behavior of the function / at zero, one gets, <4 - 2 -6 9 > V ° r cos 6 ) for S sufficiently close to 1. Thus, if there exists constant c > 0 so that , 90 (4.2.71) (1 - <5)« “ costf4" ^ 1 ^ a° ° V °S c( 1 - 5)a cosO for 5 sufficiently close to 1. This is a contradiction as the second part of the latter inequality tends to zero as S tends to 1. This proves pointwise convergence of the norm to zero. Uniform convergence is an easy consequence of convexity. □ E x a m p l e 4.2.72. There are bodies K and L in R2 so that the equation (4.2.38) defines the zero seminorm for which a different normalization gives a limiting convolution body equal to the two dimensional cube. C onstruction . We use the same construction of the last example but now using the function, for x > 0 for x < 0. Assume (® 1, 3:2) is a point on the boundary of C(5;K,L) with 0 < x\ < x2. By symmetry, it is enough to show that x% after proper normalization, converges to 1 (i.e., it is independent of x{). The same computation as in the previous example leads to the following in equalities: (4.2.73) e~ *2 + e~ * 2—/t*i) < (1 — 5)vol(AT n L) < 2e~ *2 . 91 The righthandside inequality leads easily to, (4.2.74) liminf X2 , > 1. * - i- 1/ log^s From the lefthandside inequality, after factoring out the quantity e -lj/x2 and using the monotonicity of / and its behavior at zero, we get, (4.2.75) 1 0 g 4 ( 1 — (S) where g{5) is a quantity tending to 2 as 6 tends to 1. 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Wittgenstein, L., Remarks on the foundations of Mathematics (Wright von, G. H., Rhees, R. and Anscombe, G. E. M., eds.), MIT Press, 1991. 0) we have: