A Multivariate Gnedenko Law of Large Numbers Has Also Been Considered by Goodman [12] in the Setting of Gaussian Measures on Separable Banach Spaces
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We define 0/0= to allow for the trivial case when µ has bounded support. The condition (1∞) implies super-exponential decay of the tail probabilities 1 F (t), that is, for all c> 0, − lim ect(1 F (t)) = 0. t→∞ − The converse is almost true and can be achieved if we impose some sort of regularity on F . One such regularity condition is log-concavity; see Section 3. Of course all of this can be re-worded in multiplicative form. Provided 1 F (t) is regular enough and decays super-polynomially, that is, for any m N−, ∈ lim tm(1 F (t)) = 0, t→∞ − then (2) and (3) hold, and with probability , Pn max γ n { i}1 1 δ . T − ≤ n n n Note that rapid decay of the left-hand tail provides concentration of min γi 1 , n n n { } and that [min γi 1 , max γi 1 ] = conv γi 1 . In this paper{ we} extend{ the} Gnedenko{ } law of large numbers to the multi- n variate setting. We consider a large collection of i.i.d. random vectors xi 1 in Rd that follow a log-concave distribution µ with density function f{. The} object of interest is the convex hull P = conv x n, which is called a ran- n { i}1 dom polytope. It is shown that with high probability, Pn approximates a deterministic body F1/n called the floating body, which is what remains of Rd after deleting all open half-spaces H such that µ(H) < 1/n. As in the one- dimensional case, the way in which Pn approximates F1/n depends on how rapidly µ decays. Of primary interest is a quantitative analysis in terms of the number of points, and in this regard our results are essentially optimal; see Section 8. The fact that the floating body can be used in order to model random polytopes is well known in the setting where µ is the uniform distribu- tion on a convex body; see, for example, [4] and [3]. Our main contribution is to study this approximation in the more general setting of log-concave measures. Unlike the former case, the objects that we study can have many different shapes as n and are not limited to lie within a bounded region of space. →∞ The notion of a multivariate Gnedenko law of large numbers has also been considered by Goodman [12] in the setting of Gaussian measures on separable Banach spaces. In his paper he shows that with probability 1, the n Hausdorff distance between the sample xi 1 and the ellipsoid √2 log n converges to zero as n , where is{ the} unit ball of the reproducingE kernel Hilbert space associated→∞ to µ. E LAW OF LARGE NUMBERS 3
2. Main results. Let d 1, n d + 1 and let µ be a log-concave prob- ability measure on Rd with≥ a density≥ function f = dµ/dx. This means that n f is of the form f(x) = exp( g(x)) where g is convex. Let (xi)1 denote a sequence of i.i.d. random vectors− in Rd with distribution µ, and consider the random polytope P = conv x n. For any x Rd, define n { i}1 ∈ f(x)=inf µ(H), H where H runs through the collectione of all open half-spaces that contain x. For any δ> 0, the floating body is defined as (4) F = x Rd : f(x) δ . δ { ∈ ≥ } Note that Fδ is the intersection of all closede half-spaces H such that µ(H) 1 δ, and is therefore convex. If H is any open half-space that contains≥ the− centroid of µ, then µ(H) e−1 (see Lemma 5.12 in [16] or Lemma 3.3 ≥ −1 in [8]); hence Fδ is nonempty provided that δ e . Such a floating body was defined by Sch¨utt and Werner [24] in the≤ case where µ is the uniform distribution on a convex body. We define the logarithmic Hausdorff distance between convex bodies K,L Rd as ⊂ −1 dL(K,L)=inf λ 1 : x int(K L), λ (L x)+ x K λ(L x)+ x , { ≥ ∃ ∈ ∩ − ⊂ ⊂ − } where we use the convention that inf(∅)= . The main result of the paper is as follows: ∞
Theorem 1. There exist universal constants c, c′, c> 0 with the follow- ing property. Let q 1, d N and n c exp exp(5d)+ c′ exp(q3). Let µ be ≥ R∈d ≥ e n a probability measure on with a log-concave density function, (xi)1 an i.i.d. sample from µ, P = conv x n and F the floating body as in (4). n { i}1 1/n With probability at least 1 3d+3(log n)−q, − log log n (5) dL(P , F ) 1+ cd(d + q) . n 1/n ≤ log n e The strategy of the proof is to use quantitative bounds in the one-dimen- sional case to analyze the dual Minkowski functional of Pn in different direc- tions. The idea is simple; however, there are some subtle complications. The lack of symmetry is a complicating factor, and the fact that the half-spaces of mass 1/n do not necessarily touch F1/n adds to the intricacy of the proof. We define f to be p-log-concave if it is of the form f(x)= c exp( g(x)p) where g is a nonnegative convex function and c> 0. −
Theorem 2. For all q> 0, p> 1 and d N, and any probability mea- sure µ on Rd with a p-log-concave density∈ function, there exist c, c> 0 N n such that for all n with n d + 2, if (xi)1 is an i.i.d. sample from µ, ∈ ≥ e 4 D. FRESEN
P = conv x n and F is the floating body as in (4), then with probability n { i}1 1/n at least 1 c(log n)−q we have − log log n (6) e dH(Pn, F ) c . 1/n ≤ (log n)1−1/p Theorem 2 can easily be extended to a much larger class of log-concave distributions. Using Theorem 1, any bound on the growth rate of diam(F1/n) automatically transfers to a bound on dH(Pn, F1/n). Our prototypical example is the class of distributions introduced by Schecht- d p man and Zinn [22] of the form f(x)= cp exp( x p), where 1 p< and −1 −k k d 1/p d≤ ∞ cp = p/(2Γ(p )). For these distributions, Pn (log(cpn)) Bp . Of particu- lar interest is the Gaussian distribution, where≈ p = 2. In this case (actually for the standard Gaussian distribution), B´ar´any and Vu [5] obtained a sim- ilar approximation (see Remark 9.6 in their paper) and showed that there exist two radii, R and r, both functions of n and d, such that for any fixed d 2 both r, R = (2 log n)1/2(1 + o(1)) as n , and with “high proba- ≥ d d →∞ bility” rB2 Pn RB2 . Their sandwiching result served as a key step in their proof of⊂ the⊂ central limit theorem for Gaussian polytopes (asymptotic normality of various functionals such as the volume and the number of faces). In the setting where µ is the uniform distribution on a convex body, the floating body is usually denoted by Kδ. In this context it is trivial that limn→∞ dH(Pn, K) = 0 (almost surely), and the phenomenon of interest is the rate at which Pn approached the boundary of K. B´ar´any and Larman [4] proved that for n n (d), ≥ 0 c′ vol (K K ) Evol (K P ) c′′(d) vol (K K ). d \ 1/n ≤ d \ n ≤ d \ 1/n The reader may be interested to contrast our results with the results in [9]. The results presented here require a very large sample size and guarantee a precise approximation, somewhat in the spirit of the “almost-isometric” theory of convex bodies. On the other hand, the results presented in [9] describe a type of approximation in the spirit of the “isomorphic” theory, and are most interesting, specifically in high-dimensional spaces. We also study two other deterministic bodies that serve as approximants to the random body. Define
♯ f (x)=inf f(y) dH(y), H ZH where runs through the collection of all hyperplanes that contain x, and d standsH for Lebesgue measure on . For any δ> 0, define the bodies H H D = Cl x Rd : f(x) δ , δ { ∈ ≥ } R = Cl x Rd : f ♯(x) δ , δ { ∈ ≥ } where Cl(E) denotes the closure of a set E. By log-concavity of f, both Dδ and Rδ are convex. LAW OF LARGE NUMBERS 5
Theorem 3. Let d N, and let µ be a probability measure on Rd with a continuous nonvanishing∈ log-concave density function. Then we have
(7) lim dL(Fδ, Dδ) = 1, δ→0
(8) lim dL(Fδ,Rδ) = 1. δ→0 Similar results hold in the Hausdorff distance for log-concave distributions that decay super-exponentially. Let X Rd be a random vector with distribution µ. The random vari- able log∈f(X) is a type of differential information content; see [7]. The differential− entropy of µ is defined as h(µ)= E log f(X) − = f(x) log f(x) dx − ZRd and the entropy power defined as N(µ) = exp(2d−1h(µ)). Note that the distribution of log f(X) can be expressed in terms of the function δ − 7→ µ(Dδ), P log f(X) t = µ(D ) : δ = e−t. {− ≤ } δ Because of the rapid decay of f, the body Dδ acts as an essential support for the measure µ. For δ = e−d, this was studied by Klartag and Milman [14]; see Lemma 2.2 and Corollary 2.4 in their paper. Bobkov and Madiman later provided a more precise description. In [7] they show that the variance of log f(X) is at most Cd, where C > 0 is a universal constant, and that in− high-dimensional spaces, f(X)2/d is strongly concentrated around N(µ). Theorem 1.1 in their paper can be written as µ x Rd : N(µ)−d/2δ
Theorem 4. For all d N, there exists a probability measure µ on Rd ∈ ∞ with the following universality property. Let (xi)1 be an i.i.d. sample from N n µ, and for each n with n d + 1, let Pn = conv xi 1 . Then with proba- bility 1, the sequence∈ (P )∞ ≥is dense in with respect{ } to d . n d+1 Kd BM
Throughout the paper we will make use of variables c, c, c1, c2, n0, m, etc. At times they represent universal constants, and at other times they depend on parameters such as the dimension d or the measure µe. Such dependence will always be clear from the context, and will either be indicated explicitly as cd, c(d), n0(d), etc., or implicitly as in Theorem 2, where c and c depend d on q, p, d and µ. Half-spaces shall be indexed as Hθ,t = x R : x,θ t and hyperplanes as = x Rd : x,θ = t , where θ {Sd∈−1 andhet i≥R. } Hθ,t { ∈ h i } ∈ ∈ 3. Background. Most of the material in this section is discussed in [1, d 2, 17] and [18]. We denote the standard Euclidean norm on R by 2. For any ε> 0, an ε-net in Sd−1 is a subset such that for any distinctk · k ω ,ω , ω ω > ε, and for all θ SNd−1 there exists ω such 1 2 ⊂N k 1 − 2k2 ∈ ∈N that θ ω 2 ε. Such a subset can easily be constructed using induction. By ak standard− k ≤ volumetric argument, we have 3 d (11) . |N|≤ ε By induction, any θ Sd−1 can be expressed as a series ∈ ∞ (12) θ = ω0 + εiωi, Xi=1 i where each ωi and 0 εi ε . To see this, express θ = ω0 + r0, where ∈N ≤ ≤ −1 d−1 ω0 and r0 2 ε. Then express r0 r0 S in a similar fashion, and∈N iterate thisk k procedure.≤ k k ∈ Define the functional x = max x,ω : ω . k kN {h i ∈ N } As an easy consequence of the Cauchy–Schwarz inequality, provided ε (0, 1) we have ∈ (13) (1 ε) x x x , − k k2 ≤ k kN ≤ k k2 which implies that (14) Bd B (1 ε)−1Bd, 2 ⊂ N ⊂ − 2 where BN = x : x N 1 . The body BN is what remains if one deletes all open half-spaces{ k k that≤ are} tangent to Bd at points in . 2 N LAW OF LARGE NUMBERS 7
A convex body is a compact convex subset of Euclidean space with nonempty interior. For a convex body K Rd that contains the origin as an interior point, its Minkowski functional⊂is defined as x = inf λ> 0 : x λK k kK { ∈ } d for all x R . By convexity of K, one can easily show that K obeys the triangle inequality.∈ The dual Minkowski functional is definedk · as k
y ◦ = sup x,y : x K k kK {h i ∈ } for all y Rd, and the polar of K is ∈ ◦ d K = y R : y ◦ 1 . { ∈ k kK ≤ } By the Hahn–Banach theorem, K◦◦ = K. The Hausdorff distance dH between K and L is defined as
dH(K,L)=max max d(k,L); max d(K, l) . n k∈K l∈L o By convexity this reduces to
dH(K,L)= sup sup k,θ sup l,θ θ∈Sd−1 k∈Kh i− l∈L h i
= sup ( θ K◦ θ L◦ ) . θ∈Sd−1| k k − k k | We define the logarithmic Hausdorff distance between K and L about a point x int(K L) as ∈ ∩ −1 dL(K,L,x)=inf λ 1 : λ (L x)+ x K λ(L x)+ x { ≥ − ⊂ ⊂ − } provided int(K L) = ∅, and ∩ 6 dL(K,L)=inf dL(K,L,x) : x int(K L) . { ∈ ∩ } Note that
log dL(K,L, 0)= sup log θ K log θ L . θ∈Sd−1| k k − k k | The following relations follow from the definitions above, ◦ ◦ dL(K,L, 0) = dL(K ,L , 0), (15) dL(TK,TL)= dL(K,L), where T is any invertible affine transformation. In addition, one can check that 2 dBM(K,L) dL(K,L) , (16) ≤ d (K,L) diam(K)(dL(K,L) 1); H ≤ − hence all of our bounds in terms of dL apply equally well to dBM. For large d bodies, d is more sensitive than dL. More precisely, if r 1 and rB +x K H ≥ 2 ⊂ 8 D. FRESEN for some x Rd, and if d (K,L) 1/2, then ∈ H ≤ −1 (17) dL(K,L) 1 + 2r d (K,L). ≤ H By a simple compactness argument, there is an ellipsoid of maximal vol- ume K K. This ellipsoid is called the John ellipsoid [1] associated to K. It can beE shown⊂ that is unique and has the property that K d( x)+ x, Ek ⊂ Ek − where x is the center of k. In particular, dL( k, K) d. In [10] it is shown thatE provided λ< 8−d, weE have≤ 1/d (18) dL(K, K ,x) 1 + 8λ , λ ≤ where x is the centroid of K and Kδ is the floating body inside K. The cone measure on ∂K is defined as µ (E) = vol ( rθ : θ E, r [0, 1] ) K d { ∈ ∈ } for all measurable E ∂K. The significance of the cone measure is that it leads to a natural polar⊂ integration formula (see [19]); for all f L (Rd), ∈ 1 ∞ d−1 (19) f(x) dx = d r f(rθ) dµK(θ) dr. ZRd Z0 Z∂K A probability measure µ is called isotropic if its centroid lies at the origin and its covariance matrix is the d d identity matrix. A function f : Rd [0, ) is called× log-concave (see [14]) if → ∞ f(λx + (1 λ)y) f(x)λf(y)1−λ − ≥ for all x,y Rd and all λ (0, 1). Any such function can be written in the form f(x)=∈ e−g(x) where g∈: Rd ( , ] is convex. If f is the density of a probability measure µ, then it→ must−∞ decay∞ exponentially to zero. In this case g lies above a cone, that is, (20) g(x) m x c ≥ k k2 − with m,c> 0. As a consequence of the Pr´ekopa–Leindler inequality [2], if x is a random vector with log-concave density, and y is any fixed vector, then x,y has a log-concave density in R. Log-concave functions are very rigid. hOnei such example of this rigidity (see Lemma 5.12 in [16]) is the fact that if H is any half-space containing the centroid of µ, then µ(H) e−1. Another example (see Theorem 5.14 in [16]) is that if µ is isotropic, then≥ (21) 2−7d f(0) d(20d)d/2, ≤ ≤ (22) (4πe)−d/2 f 28ddd/2, ≤ k k∞ ≤ and if x 1/9, then k k2 ≤ (23) 2−8d f(x) d2d(20d)d/2. ≤ ≤ LAW OF LARGE NUMBERS 9
d Let 1 p< . If g : R [0, ] is convex and limx→∞ g(x)= , then the probability≤ measure∞ with→ density∞ given by ∞ p f(x)= ce−g(x) will be called p-log-concave. This is a natural generalization of the normal distribution. If f is p-log-concave, then it is also p′-log-concave for all 1 p′ p. ≤ ≤ Let Hd denote the collection of all (d 1)-dimensional affine subspaces (hyperplanes) of Rd. The Radon transform− of an integrable log-concave func- tion f : Rd [0, ) is the function Rf : H [0, ) defined by → ∞ d → ∞ (24) Rf( )= f(y) dH(y), H ZH where dH is Lebesgue measure on . The Radon transform is closely related to the Fourier transform. See [15] forH a discussion of these operators and their connections to convex geometry.
4. The one-dimensional case. Let f be a nonvanishing log-concave prob- ability density function on R associated to a probability measure µ. In par- ticular, f(t)= e−g(t) where g : R R is convex. For t R, define → ∈ t J(t)= f(s) ds, Z−∞ u(t)= log(1 J(t)). − − The cumulative distribution function J is a strictly increasing bijection be- tween R and (0, 1). The following lemma is a standard result; see, for exam- ple, Theorem 5.1 in [16] for the statement, and the references given there. However, we include a short proof here for completeness.
Lemma 5. u is convex.
Proof. Assume momentarily that g C2(R). For t (0, 1) define ∈ ∈ ψ(t)= f(J −1(1 t)). − Note that g′′(J −1(1 t)) ψ′′(t)= − − 0. ψ(t) ≤
Hence ψ is concave. In addition, limt→0 ψ(t) = limt→1 ψ(t) = 0. Hence, the function κ(t)= ψ(t)/t is nonincreasing on (0, 1), and the function f(t)/(1 J(t)) = κ(1 J(t)) is nondecreasing on R. Since u′(t)= f(t)/(1 J(t)), u −is convex. − − If g / C2(R), then the result follows by approximation (convolve µ with a Gaussian).∈
Lemma 6. If T 1 and x> 2T log T , then (log x)/x < T −1. ≥ 10 D. FRESEN
Proof. Since the function y = e−1x is tangent to the strictly concave function y = log x, the function y = (log x)/x has a global maximum of e−1 and is decreasing on [e, ). We now consider two cases. In case 1, T < e and therefore (log x)/x e−∞1 < T −1. In case 2, T e. Since (log T )/T < 2−1, it follows that log(2 log≤ T ) < log T . For x′ = 2T log≥ T , log x′ log T + log(2 log T ) 1 = < . x′ 2T log T T Since x>x′ > e, (log x)/x < (log x′)/x′ and the result follows.
The following lemma is a quantitative version of the Gnedenko law of large numbers for log-concave probability measures on R.
Lemma 7. Let q 1 and n 120q2(2 + log q)2. Let µ be a probability measure on R with a≥ nonvanishing≥ log-concave density function and cumu- n lative distribution function J, and let (γi)1 be an i.i.d. sample from µ. With probability at least 1 2(log n)−q, − −1 γ(n) J (1 1/n) log log n (25) | − − | 6q , J −1(1 1/n) Eµ ≤ log n − − where γ = max γ n and Eµ denotes the centroid of µ. (n) { i}1 Proof. We shall implicitly make use of Lemma 6 several times through- out the proof. Let a = (log n)−q and b = q log n. It follows that 0 −1 −1 If the event J (1 b/n) γ(n) J (1 a/n) occurs, then the event defined by inequality{ − (25) also≤ occurs.≤ − } Although Lemma 7 applies to the multiplicative version of the Gnedenko law of large numbers, it also recovers the additive version as long as log n (27) J −1(1 1/n)= o . − log log n −1 If, in the proof, we take a = b = log(m) n (the mth iterate of the logarithm), −1 then the probability bound becomes 1 2(log(m) n) , and the right-hand side of (25) becomes − 4 log(m+1) n log n provided n>n0(m). 5. Main proofs. Since Lebesgue measure depends on the underlying Eu- clidean structure of Rd, so does the definition of f = dµ/dx, and therefore also the definition of D = Cl x : f(x) δ . A natural variation of the body δ { ≥ } Dδ which does not depend on Euclidean structure is the body D♮ = Cl x Rd : f(x) τ −19d det cov(µ) −1/2δ , δ { ∈ ≥ d | | } where the quantity 1 d−1 2 (d−1)/2 (28) τd = vold−1(B2 ) (1 t ) dt Z1/2 − d represents the volume of the set x R : x 2 1, x1 1/2 . Associated ♮ { ∈ k k ≤ ≥ } to Dδ are three ellipsoids that play a central role in our proof. The John ♮ ellipsoid of Dδ is denoted D♮ and the centroid of D♮ will be denoted δ. E δ E δ O We also consider ♯ (29) δ = 3d( D♮ δ)+ δ E E δ − O O and ♭ 1 (30) δ = 2 ( D♮ δ)+ δ. E E δ − O O ♮ The advantage of using Dδ is that we may place µ in different positions at various stages of our analysis. We first position µ to be isotropic and then d position it so that D♮ = B2 . We include the proofs of Lemmas 8 and 9 in E δ Section 6. Lemma 8. There exists a universal constant c> 0 with the following property. Let d N, let µ be a log-concave probability measure with a contin- uous density function∈ f, and let δ < c exp( 8d2 log d). Let H be a half-space − (either open or closed) with µ(H)= δ, and let ♯ and ♭ be defined by (29) Eδ Eδ 12 D. FRESEN and (30), respectively. Then (31) H ♯ = ∅, ∩Eδ 6 (32) H ♭ = ∅. ∩Eδ Consequently, ♭ F ♯. Eδ ⊂ δ ⊂Eδ We shall use the Euclidean structure corresponding to D♮ in order to E δ compare F1/n and Pn. The following lemma together with Lemma 8 allows us to do so. Lemma 9. Let d N and let K and L be convex bodies in Rd such ∈d d that 0 int(L) and rB2 K RB2 for some r,R> 0. Let 0 <ρ< 1/2 and 0 <ε<∈(16R/r)−1, and⊂ let ⊂ be an ε-net in Sd−1. Suppose that for each ω , N ∈N (33) (1 ρ) ω ω (1 + ρ) ω . − k kL ≤ k kK ≤ k kL Then for all x Rd we have ∈ (34) (1 + 2ρ + 28Rr−1ε)−1 x x (1+2ρ + 28Rr−1ε) x . k kL ≤ k kK ≤ k kL In particular, −1 (35) dL(K,L) dL(K,L, 0) 1 + 2ρ + 28Rr ε. ≤ ≤ Proof of Theorem 1. By convolving µ with a Gaussian measure of the form −d −1 φλ,d(x)= λ φd(λ x), −d −1 2 where φd(x) = (2π) exp( 2 x 2) is the standard normal density func- tion, and taking λ 0, we− mayk assumek that the density of µ is continuous and nonvanishing. This→ is possible because the bounds in the theorem do not depend on µ. The condition n c exp exp(5d)+ c′ exp(q3) (with sufficiently large c and c′) insures that the≥ probability bound is nontrivial. It is also sufficiently large so that we may use Lemma 8 with δ = 1/n and Lemma 7 with q′ = d + q. In fact we will implicitly make use of this bound repeat- edly throughout the proof. Let ε = (log n)−1. By applying a suitable affine d H transformation, we may assume that D♮ = B2 . By Lemma 8, if θ,t is a E 1/n half-space with µ(Hθ,t) = 1/n, then (36) 1/2 t 3d. ≤ ≤ This implies that 1/2Bd F 3dBd. For each θ Sd−1, the function 2 ⊂ 1/n ⊂ 2 ∈ f (t)= d µ(H ) is the density of a log-concave probability measure µ on θ − dt θ,t θ R with cumulative distribution function Jθ(t) = 1 µ(Hθ,t). Furthermore, the sequence ( θ,x )n is an i.i.d. sample from this− distribution. Recalling h ii i=1 LAW OF LARGE NUMBERS 13 the definition of the dual Minkowski functional, for any y Rd, ∈ y ◦ = sup x,y : x P k kPn {h i ∈ n} = max xi,y . i=1,...,nh i We use this notation even when 0 / int(Pn). Let denote a generic ε-net in Sd−1, and consider the function ∈ N f (x)=inf µ(H ) : ω ,t = ω,x . N { ω,t ∈N h i} For all δ> 0, definee the discrete floating body F N = x Rd : f (x) δ . δ { ∈ N ≥ } e N Note that f(x)=infN fN (x) and Fδ = N Fδ , where runs through the d−1 1 d N N collection of all ε-nets in S . By (36), TB F 3dBN , and by (14) we e e 2 2 ⊂ 1/n ⊂ have 1/2Bd F N 4dBd which implies that (4d)−1Bd (F N )◦ 2Bd. 2 ⊂ 1/n ⊂ 2 2 ⊂ 1/n ⊂ 2 For each θ Sd−1, we have ∈ Eµ J −1(e−1) θ ≥ θ J −1(1/n). ≥ θ Combining this and (25), with probability at least 1 2(log n)−d−q we have that − −1 θ P ◦ J (1 1/n) log log n |k k n − θ − | 6(d + q) . J −1(1 1/n) J −1(1/n) ≤ log n θ − − θ −1 −1 Since both Jθ (1/n) and Jθ (1 1/n) lie in the interval [1/2, 3d], both have roughly− the same order of magnitude,− and we have −1 −1 (1 ρ)J (1 1/n) θ ◦ (1 + ρ)J (1 1/n), − θ − ≤ k kPn ≤ θ − where log log n ρ = 42d(d + q) < 1/8. log n With probability at least 1 ε−d3d+1(log n)−d−q = 1 3d+1(log n)−q, this holds for all ω . Hence, − − ∈N (1 + ρ)−1P F N , n ⊂ 1/n which implies that (1 ρ) θ P ◦ θ (F N )◦ − k k n ≤ k k 1/n 14 D. FRESEN for all θ Sd−1. On the other hand, for all ω we have ∈ ∈N −1 ◦ ω Pn (1 ρ)Jω (1 1/n) (37) k k ≥ − − (1 ρ) ω (F N )◦ ≥ − k k 1/n As ◦ is the supremum of Lipschitz functions, Lip( ◦ ) sup x : k · kPn k · kPn ≤ {k k2 x P < 8d. Since (37) implies that ω ◦ > 1/4 (simultaneously for all ∈ n} k kPn d−1 ◦ ω with high probability), for all θ S we have θ Pn > 1/8. Using ∈N ∈ k k ◦ the Hahn–Banach theorem, 0 int(Pn). This also implies that 0 int(Pn ). By (34), ∈ ∈ −1 (38) (1 + 4ρ + 224dε) x P ◦ x (F N )◦ (1 + 4ρ + 224dε) x P ◦ k k n ≤ k k 1/n ≤ k k n for all x Rd. Let be any other ε-net in Sd−1. By the calculations above, with probability∈ atM least 1 3d+1(log n)−q, − −1 (39) (1 + 4ρ + 224dε) x P ◦ x (F M )◦ (1 + 4ρ + 224dε) x P ◦ k k n ≤ k k 1/n ≤ k k n for all x Rd. By the union bound, with probability at least 1 3d+2(log n)∈−q > 0, both (38) and (39) hold, which implies that − (40) (1 + 4ρ + 224dε)−2F N F M (1 + 4ρ + 224dε)2F N . 1/n ⊂ 1/n ⊂ 1/n N M However both F1/n and F1/n are deterministic bodies, and (40) therefore M holds unconditionally. Since F1/n = M F1/n, where the intersection is taken over all ε-nets in Sd−1, we have T (1 + 4ρ + 224dε)−2F N F (1 + 4ρ + 224dε)2F N . 1/n ⊂ 1/n ⊂ 1/n Combining this with the polar of (38) gives that with probability at least 1 3d+1(log n)−q, we have − (1 + 4ρ + 224dε)−3P F (1 + 4ρ + 224dε)3P n ⊂ 1/n ⊂ n from which the result follows by the inequality (1 + ε′)3 1 + 12ε′, valid if 0 ε′ 1. ≤ ≤ ≤ d Lemma 10. Let g : R [0, ] be convex with limx→∞ g(x)= , let K Rd be a convex body containing→ ∞ 0 in its interior and let p> 1.∞ Then there⊂ exist c , c > 0 such that for all x Rd, 1 2 ∈ (41) g(x)p c x p c . ≥ 1k kK − 2 Proof. We leave the easy proof of this to the reader. Lemma 11. Let p> 1, d N and let µ be a p-log-concave probability measure on Rd. Then there exist∈ c , c ,t > 0 such that for all θ Sd−1 and 1 2 0 ∈ LAW OF LARGE NUMBERS 15 all t t0, ≥ p (42) µ(H ) c t1−pe−c2t , θ,t ≤ 1 where H = x Rd : x,θ t . θ,t { ∈ h i≥ } Proof. For all t 1 we have ≥ p d p e−t (p−1t1−pe−t ) ≤−dt p p = p−1(p 1)t−pe−t + e−t − p p−1(2p 1)e−t . ≤ − Hence, by the fundamental theorem of calculus, ∞ p p p (43) (2p 1)−1t1−pe−t e−s ds p−1t1−pe−t . − ≤ Zt ≤ Since the image of a p-log-concave probability measure under an orthogonal transformation is p-log-concave, we may assume without loss of generality that θ = e1 = (1, 0, 0, 0,...). By (41), there exist c1, c2 > 0 such that for all x Rd, ∈ p f(x) c e−c2kxkp , ≤ 1 p d p where x p = x . Hence, k k i=1 | i| P p −c2kxkp µ(Hθ,t) c1e dx ≤ ZHθ,t (44) ∞ p −c2s = c3e ds. Zt The result now follows from a change of variables, (44) and (43). Proof of Theorem 2. Let c1, c2 and t0 be the constants appearing −1 p in Lemma 11. Let n0 > c1 + exp(2 c2t0). Without loss of generality, t0 > 1 −1 1/p Rd and n>n0. Set α = (2c2 log n) and consider any x with x 2 > α. −1 ∈ k k Let θ = x 2 x and t = (α+ x 2)/2. Since t>α>t0 and n > c1, Lemma 11 implies thatk k k k −2 −1 µ(Hθ,t) < c1n Since x 2 >t, x int(Hθ,t). By definition of the floating body, x / F1/n. k k ∈ 1/p ∈ Since this is true for all such x, diam(F1/n) 2α = c4(log n) . The result now follows from Theorem 1 and relation (16≤) between the Hausdorff and the logarithmic Hausdorff distances. 6. Technical lemmas. This section contains some technical results on the rigidity of log-concave functions that enable us to obtain a lower bound on the sample size. 16 D. FRESEN Lemma 12. There exist universal constants c1, c2 > 0 such that for all d N, ∈ cdd−d/2 vol (Bd) cdd−d/2. 1 ≤ d 2 ≤ 2 d Proof. This follows from Stirling’s formula and the expression vold(B2 )= πd/2(Γ(1 + d/2))−1; see Corollary 2.20 in [15] or page 11 in [20]. Lemma 13. There exists a universal constant c> 0 with the following property. Let d N and let µ be an isotropic log-concave probability measure on Rd with density∈ function f. For all x Rd, ∈ f(x) e−αdkxk2+βd , ≤ d −d/2 where αd = c d and βd = 10d log(d) + 7. Proof. We first consider the case d 2. The volume of a cone in Rd −1 d−≥1 d−1 Rd with height h and base radius r is d r hvold−1(B2 ). For any x , d ⊥ ∈ let Ax be the cone with vertex x and base (1/9)B2 x . Then vold(Ax)= −1 −d+1 d−1 −4d+3 d−∩1 d 9 x 2 vold−1(B2 ) > e x 2 vold−1(B2 ). By log-concavity of f and inequalityk k (23), for all y A ,k k ∈ x (45) f(y) min f(x), 2−8d . ≥ { } If f(x) 2−8d, then ≥ −8d −10d+3 d−1 1 f(y) dy 2 vold(Ax) > e x 2 vold−1(B2 ), ≥ ZAx ≥ k k and it follows that e10d−3 x 2 < d−1 . k k vold−1(B2 ) Hence, if x e10d−3/ vol (Bd−1), then k k2 ≥ d−1 2 (46) f(x) < 2−8d. For any such x we have the convex combination e10d−3 x e10d−3 e10d−3 d−1 = d−1 x + 1 d−1 0. vol (B ) x 2 x vol (B ) − x vol (B ) d−1 2 k k k k2 d−1 2 k k2 d−1 2 Set e10d−3 x x = d−1 . vol (B ) x 2 d−1 2 k k Using concavity of log f ande inequality (46), e10d−3 e10d−3 8d ln 2 log f(x)+ 1 log f(0). − ≥ x vol (Bd−1) − x vol (Bd−1) k k2 d−1 2 k k2 d−1 2 LAW OF LARGE NUMBERS 17 After some simplification, and using inequality (21), we get f(x) exp( de−10d+3vol (Bd−1) x ln 2 7d ln 2). ≤ − d−1 2 k k2 − If, on the other hand, x < e10d−3/vol (Bd−1), then by (22) k k2 d−1 2 f(x) f dd/228d ≤ k k∞ ≤ = exp(2−1d ln d + 8d ln 2). Using Lemma 12, it follows that for all x Rd, ∈ f(x) exp( de−10d+3 vol (Bd−1) ln 2 x + 9d ln 2 + 2−1d ln d) ≤ − d−1 2 k k2 exp( cdd−d/2 x + 10d ln d). ≤ − 3 k k2 The case d = 1 is simpler, and we leave the details to the reader. First show that f(28) 2−8, and then proceed as in the case d 2 to obtain f(x) exp( 2−9 ≤x + 7) for all x R. ≥ ≤ − | | ∈ Corollary 14. There exist universal constants c1, c2 > 0 with the fol- lowing property. Let d N, and let µ be an absolutely continuous isotropic log-concave probability∈ measure. For all δ < e−10d log d−7, D cddd/2(log δ−1)Bd. δ ⊂ 1 2 In particular, vol (D ) c exp(d2 log d)(log δ−1)d. d δ ≤ 2 Proof. By (21), Dδ = ∅. By the bounds on δ, it follows that 10d log d + 7 log δ−1. The result now6 follows from Lemmas 13 and 12. ≤ Lemma 15. There exists a universal constant c> 0 with the following property. Let d N, and let µ be an isotropic log-concave probability measure with density f.∈ Let r> 1 and x Rd. If f(x) < 2−8d, then ∈ (47) f(rx) f(x) exp( cdd−d/2(r 1) x ). ≤ − − k k2 Proof. Let g = log f. By Lemmas 13 and 12, there exists a universal − −8d d d/2 constant c2 > 0 such that f(x) 2 for all x with x 2 c2d ; see in par- ticular (46). Let x Rd be the point≤ specified in thek statementk ≥ of the lemma. ∈ e e d d/2e d d/2 −1 We consider two cases. In the first case x 2 c2d . Let x = c2d x 2 x. −7d k k ≥ k k By inequality (21), f(0) 2 . By convexity of g and the definition of c2, ≥ e g(rx) g(x) g(x) g(0) −1 f(0) −d 1−d/2 − − = x 2 ln c2 d ln 2. (r 1) x 2 ≥ x 2 k k f(x) ≥ − k k ek k d d/2 e −8d In the second case, x 2 < ce2d . Recall that, bye hypothesis, f(x) < 2 . Therefore, k k g(rx) g(x) g(x) g(0) f(0) − − x −1 ln c−dd1−d/2 ln(2) (r 1) x ≥ x ≥ k k2 f(x) ≥ 2 − k k2 k k2 −1 from which the result follows with c = (2c2) . 18 D. FRESEN Lemma 16. There exists a universal constant c1 > 0 with the following property. Let d N, and let µ be an isotropic log-concave probability measure with a continuous∈ density function f. For all δ < e−10d log d−7, (48) µ(Rd D ) α δ(log δ−1)d, \ δ ≤ d 2 where αd = c1 exp(3d log d). Proof. Since f is continuous, for all θ ∂Dδ we have f(θ)= δ. By the polar integration formula (19) and inequality∈ (47), d µ(R Dδ)= f(x) dx d \ ZR \Dδ ∞ d−1 = d r f(rθ) dµDδ (θ) dr Z1 Z∂Dδ ∞ d−1 d −d/2 d r δ exp( c d (r 1) θ 2) dµDδ (θ) dr. ≤ Z1 Z∂Dδ − − k k By (23) and the fact that δ< 2−8d, we have 1/9Bd D . By Corollary 14, 2 ⊂ δ ∞ Rd d−1 d −d/2 −1 µ( Dδ) d r δ exp( c d (r 1)9 ) dµDδ (θ) dr \ ≤ Z1 Z∂Dδ − − ∞ d−1 d −d/2 δ vold(Dδ)d r exp( c2d (r 1)) dr ≤ Z1 − − β δ(log δ−1)dd exp(d2 log d + c ), ≤ d 3 where ∞ d−1 d −d/2 βd = r exp( c2d (r 1)) dr. Z1 − − d −d/2 Set ωd = c2d and t = ωdr. Recall the definition of the gamma function ∞ −r z−1 Γ(z)= 0 e r dr. By a change of variables and Stirling’s formula, R ∞ d−1 βd exp(ωd) r exp( ωdr) dr ≤ Z0 − ∞ −d d−1 −t c4ωd t e dt ≤ Z0 c exp(d2 log d) ≤ 5 from which the result follows. Lemma 16 is optimal in δ up to a factor log δ−1 as can be seen from the example f(x)= c exp( x ), in which case µ(Rd D ) c δ(log δ−1)d−1 d −k k2 \ δ ≥ d for δ < δ0(d). To see this, apply the polar integration formula as in the proof e LAW OF LARGE NUMBERS 19 of Lemma 16 and use the equation ∞ d−1 −r d−1 −R r e dr =(1+ od(1))R e ZR with d fixed and R . →∞ Lemma 17. There exists a universal constant c> 0 such that for all d N, if t > d5d, then √t c(log t)d. ∈ ≥ Note that the inequality fails for t = d2d. Proof of Lemma 17. First, consider any d> 16. For any such d, (2d)4d < d5d. Set T = 2d and x = log t. Since (2d)4d < d5d Lemma 18. There exists a universal constant c> 0 with the following property. Let d N and let µ be an isotropic log-concave probability measure with a continuous∈ density function f. For all δ< c exp(e 8d2 log d), − d µ(R 2D −1 d ) < δ, \ τ 9 δ e where τ = τ = vol (Bd−1) 1 (1 t2)(d−1)/2 dt. d d−1 2 1/2 − R 2 Proof. Consider the quantity αd = c1 exp(3d log d) from Lemma 16. By concavity, 1 t2 3(1 t)/2 for all 1/2 t 3/4. By a change of vari- − ≥ − d −≤d/2≤ −1 d ables and Lemma 12, one sees that τ > c2d . Let κ = τ 9 δ. Consider any y ∂(2Dκ). Then x = y/2 ∂Dκ, and we have the convex combination x = 1 0+∈ 1 y. By log-concavity, f∈(x) f(0)1/2f(y)1/2 and by inequality (21), 2 2 ≥ f(x)2 f(y) < 28dκ2 ≤ f(0) and y / D with ε = 28dκ2. Since this is true for all y ∂(2D ), D ∈ ε ∈ κ ε ⊂ 2Dκ. For a sufficiently small choice of c (chosen independently of d), ε< e 20 D. FRESEN e−10d log d−7. By Lemma 16 and the inequality e9d 28d92d e10d, ≤ ≤ µ(Rd 2D ) µ(Rd D ) α ε(log ε−1)d \ κ ≤ \ ε ≤ d e10dα τ −2δ2(2 log δ−1 + log(e−8dτ 2))d ≤ d δ(c−1δ1/2α 2de10dτ −2)cδ1/2(log δ−1)d, ≤ d where c is the constant from Lemma 17. By the bound imposed on δ, −1 1/2 d 10d −2 c δ αd2 e τ < 1. The result now follows from Lemma 17. Recall that K denotes the John ellipsoid of a convex body K and that K d( E x )+ x , where x is the center of . EK ⊂ ⊂ EK − 0 0 0 EK Lemma 19. Let K Rd be a convex body with 0 K. Then 2K 3d( x )+ x . ⊂ ∈ ⊂ EK − 0 0 Proof. By applying a suitable linear transformation, we may assume that = Bd + x . Take any x K. Since max x x , x d, it EK 2 0 ∈ {k 0 − k2 k 0k2}≤ follows that x 2 x0 2 + x x0 2 2d and that x0 2x 2 x0 x + x 2xk k ≤3d k. k k − k ≤ k − k ≤ k − k2 k − k2 ≤ Lemma 20. Let be an ellipsoid with centroid , and let H be a half- H E d OH 1 space with vold( ) vold(B2 ) <τd vold( ). Then and 2 ( )+ are disjoint. ∩E × E E − O O Proof. The truth of the lemma is invariant under affine transforma- tions of , and we may therefore assume that = Bd. The result now follows E E 2 from the definition of τd [see equation (28)] and the fact that τd = vold x Rd : x 1, x 1/2 . { ∈ k k2 ≤ 1 ≥ } Proof of Lemma 8. Consider τ = τd defined by (28). We may assume ♮ that µ is in isotropic position, which implies that Dδ = Dτ −19dδ. Lemmas 18 H ♯ ∅ ♮ −1 d and 19 together imply that δ = . For each x Dδ, f(x) τ 9 δ. H −1 ∩Ed 6 H ∈ ≥ H Therefore δ µ( D♮ ) τ 9 δ vold( D♮ ) which implies that vold( ≥ ∩E δ ≥ ∩E δ ∩ −d −1 d −8d D♮ ) τ9 . Since the density function f is continuous and τ 9 δ< 2 , E δ ≤ −1 d ♮ inequality (23) implies that (9 + κ)B2 Dδ for some κ> 0. Since D♮ ⊂ E δ ♮ is the unique ellipsoid of maximal volume inside Dδ, we have vold( D♮ ) > E δ 9−d vol (Bd). From the definition of ♭ and Lemma 20, we see that H ♭ = d 2 Eδ ∩Eδ ∅. Finally, the claim that ♭ F follows from the definition of F while Eδ ⊂ δ δ the claim that F ♯ follows from the Hahn–Banach theorem [any x / ♯ δ ⊂Eδ ∈Eδ H H ♯ ∅ H lies in an open half-space with δ = and therefore µ( ) < δ and x / F ]. ∩E ∈ δ LAW OF LARGE NUMBERS 21 Proof of Lemma 9. Note that 1 + ρ (1 ρ)−1 1 + 2ρ and 1 ρ −1 ≤ − ≤ d − ≤ (1 + ρ) 1 ρ/2, and the same inequalities hold for ε. Since rB2 K d ≤ − ⊂ ⊂ RB2 , we have that R−1 x x r−1 x k k2 ≤ k kK ≤ k k2 for all x Rd. Combining this with (33) gives ∈ R−1(1 + ρ)−1 ω r−1(1 ρ)−1 ≤ k kL ≤ − for all ω . Consider any θ Sd−1. Let ω be the element of that ∈N ∈ 0 N minimizes θ ω0 2, and consider the series representation (12). By the triangle inequality,k − k θ r−1(1 ρ)−1(1 ε)−1. k kL ≤ − − −1 −1 −1 d Hence x L r (1 ρ) (1 ε) x 2 for all x R . Using the triangle inequalityk k in≤ a bit of− a different− way,k k ∈ ∞ θ ω ε ω k kL ≥ k 0kL − ik ikL Xi=1 R−1(1 + ρ)−1 r−1ε(1 ε)−1(1 ρ)−1 ≥ − − − R−1/2 4r−1ε ≥ − = R−1(1 8Rr−1ε)/2 − (4R)−1, ≥ which holds since 8Rr−1ε 1/2. Thus ≤ θ ω + θ ω k kL ≤ k 0kL k − 0kL (1 ρ)−1 ω + r−1(1 ρ)−1(1 ε)−1ε ≤ − k 0kK − − (1 ρ)−1( θ + ω θ )+ r−1(1 ρ)−1(1 ε)−1ε ≤ − k kK k 0 − kK − − (1 ρ)−1 θ + r−1(1 ρ)−1ε(1 + (1 ε)−1) ≤ − k kK − − (1 ρ)−1 θ + Rr−1(1 ρ)−1ε(1 + (1 ε)−1) θ ≤ − k kK − − k kK (1 + 2ρ)(1 + 3Rr−1ε) θ ≤ k kK (1 + 2ρ + 6Rr−1ε) θ . ≤ k kK On the other hand, θ ω + θ ω k kK ≤ k 0kK k − 0kK (1 + ρ) ω + r−1ε ≤ k 0kL (1 + ρ)( θ + ω θ )+ r−1ε ≤ k kL k 0 − kL 22 D. FRESEN (1 + ρ) θ + r−1(1 + ρ)(1 ρ)−1(1 ε)−1ε + r−1ε ≤ k kL − − (1 + ρ) θ + 7r−1ε 4R θ ≤ k kL · k kL (1+ ρ + 28Rr−1ε) θ . ≤ k kL The result follows by positive homogeneity. 7. Proof of Theorem 3. Fix µ and d as in the statement of Theorem 3. Let f be the density of µ, and let g = log f. All variables in this section depend on both d and µ. − Lemma 21. There exist c, ε > 0 such that for all ε (0, ε ), 0 ∈ 0 (49) µ(Rd D ) < cε(log ε−1)d. \ ε Proof. Since µ has a log-concave density, it necessarily has a nonsin- gular covariance matrix, and there exists an affine map T such that µ′ = Tµ is isotropic. The density of µ′ is f(x) = det(T −1)f(T −1x) −1 −1 ′ and Dε = T Dεe, where εe= ε det T and Dεe = x : f(x) ε . Since µ is isotropic, we may use Lemma 16, which gives { ≥ } e e e e d ′ d e µ(R D )= µ (R De) \ ε \ ε c′ε(log ε−e1)d ≤ −1 d < cεe(log εe ) . d ′ Lemma 22. For any x R there exist c , δ0 > 0 and a function p : (0, δ0) (0, ) such that for all δ ∈(0, δ ), → ∞ ∈ 0 log log δ−1 (50) p(δ) c′ ≤ log δ−1 and (51) F (1 + p)(D x)+ x. δ ⊂ δ − Proof. Let c> 0 be the constant in (49). A brief analysis of the function −1 d t ct(log t ) shows that there exists δ0 > 0 and a function ε = ε(δ) defined 7→ −1 d implicitly for all δ (0, δ0) by the equation δ = cε(log ε ) . We can take δ0 small enough to ensure∈ that ε < δ and that log δ−1 < log ε−1 < 2 log δ−1. If we define log ε−1 log δ−1 p(δ) = 3 − , log δ−1 1+p/2 then δ < ε, and (50) holds. Since Dε is both compact and convex, for any point y / D there exists (by the Hahn–Banach theorem), a closed half- ∈ ε LAW OF LARGE NUMBERS 23 space H with y H and H D = ∅. Since H Rd D , (49) implies that ∈ ∩ ε ⊂ \ ε µ(H) < δ and by definition of Fδ, y / Fδ. This goes to show that Fδ Dε. Let d d−1 ∈ ⊂ −g (t) x R . For any θ S , consider the function fθ(t)= f(x + tθ)= e θ , t ∈R. This notation∈ differs slightly from that in the proof of Theorem 1. By continuity∈ and log-concavity, if ε is small enough, then for all θ Sd−1 there ∈ −1 is a unique v> 0 such that fθ(v)= ε; we denote this number by fθ (ε). 2 −p/2 We may assume that δ0 < min 1,f(x) . Note that 1 < δ/ε < δ and −1 −1 { } log δ + log f(x) 1/2 log δ . By convexity of gθ, for any 0 < s < v we −1 ≥ −1 −1 have s (gθ(s) gθ(0)) (v s) (gθ(v) gθ(s)). Taking v = fθ (ε) and −1 − ≤ − − s = fθ (δ), this becomes −1 −1 −1 −1 fθ (ε) fθ (δ) log ε log δ −−1 −1 − . fθ (δ) ≤ log δ + log f(x) (52) < p. Inequality (52) reduces to f −1(ε) (1 + p)f −1(δ). Since this holds for any θ ≤ θ θ Sd−1, D (1 + p)(D x)+ x, and (51) follows. ∈ ε ⊂ δ − Lemma 23. There exists δ0 > 0 such that for all δ (0, δ0) we have the relation ∈ (53) (1 + 8λ1/d)−1(D x′)+ x′ F , δ − ⊂ δ −1 ′ where λ = vold(Dδ) , and x is the centroid of Dδ. d Proof. Let δ0 be such that vold(Dδ0 ) > 8 . We use the notation (Dδ)λ for the convex floating body with parameter λ> 0 corresponding to the uniform probability measure on Dδ. If H is any half-space with µ(H) < δ, −1 then vold(H Dδ) < 1. Hence (Dδ)λ Fδ, where λ = vold(Dδ) . The result now follows∩ from inequality (18). ⊂ Lemma 24. Let K,L Rd be convex bodies such that there exist x,x′ int(K L) and 0 (55) dL(K,L) 1 + 8 dr. ≤ Proof. Since the statement of the lemma is invariant under affine trans- formations that act simultaneously on K and L, we may assume without d d d loss of generality that the John ellipsoid of K is B2 . Hence B2 K dB2 ′ d ⊂ ⊂ and x 2, x 2 d. Note also that L 3dB2 . Using these facts and manip- ulatingk k (54k) ink ≤ the obvious way, we see⊂ that both of the following relations 24 D. FRESEN hold: L K + 2drBd, ⊂ 2 K L + 4drBd. ⊂ 2 By definition of the Hausdorff distance, d (K,L) 4dr < 1/2. Since Bd H ≤ 2 ⊂ K, inequality (17) implies that dL(K,L) 1 + 8dr. ≤ Proof of equation (7). Since limδ→0 p(δ) = limδ→0 λ(δ) = 0, equa- tion (7) now follows from (51), (53) and (55). Remark 25. There is no lower bound on the growth rate of vold(Dδ); indeed the function could grow arbitrarily slowly. However in the case of the d d/p d Schechtman–Zinn distributions, vold(Dδ) = (log(cp/δ)) vold(Bp ), and we leave it to the reader to combine this with (51), (50) and (53) to obtain a quantitative upper bound on dL(Fδ, Dδ). Proof of equation (8). Let ε> 0 be given. Using the notation from the proof of Theorem 1, for any θ Sd−1 we define ∈ d f (t)= µ(H ). θ −dt θ,t This function is the density of a log-concave probability measure on R with cumulative distribution function Jθ(t) = 1 µ(Hθ,t). Using Fubini’s theorem we have − f (t)= Rf( ), θ Hθ,t where Rf is the Radon transform of f as defined by (24). For all t R, d−1 ∈ the function θ fθ(t) is continuous and nonvanishing on S . This follows using the properties7→ imposed on f, together with Lebesgue’s dominated con- d−1 vergence theorem and (20). Define α = inf fθ(0) : θ S , and note that { ∈ } d−1 α> 0. By (20) there exists t0 > 0 such that if β = sup fθ(t0) : θ S , −1 { ∈ } then β < α. Define gθ(t)= log fθ(t), and let λ = t0 (log α log β) and ∆ = max 1, λ−1 log λ−1 . By− definition of α, β and λ, for− all θ Sd−1 {−1 } ∈ we have t0 (gθ(t0) gθ(0)) λ. By convexity of gθ, if u > v t0, then − ≥ −λ≥(u−v) gθ(u) gθ(v)+ λ(u v), which translates into fθ(u) fθ(v)e . Let ≥ − d−1 −1 d ≤ δ0 < inf fθ(t0 + 1) : θ S be such that ∆ε B2 Fδ0 . Consider any δ < δ , and{ momentarily∈ fix θ} Sd−1. By definition of ⊂α and β, 0 ∈ 0 By definition of δ0 and the equation fθ( u)= f−θ(u), we have min fθ(t0 + 1),f ( t 1) > δ . By log-concavity we− have f (u) δ for all {t 1 < θ − 0 − } 0 θ ≥ 0 − 0 − u Since dL is preserved by invertible affine transformations [as per (15)], the same inequality holds for all Euclidean balls. This gives (58). We can choose q to be arbitrarily small, in which case (57) and (58) complement Theorems 1 and 2. 9. Proof of Theorem 4. Let d denote the collection of all convex bodies Rd K ♯ in (compact convex sets with nonempty interior), and let d = d 0 . If Ω is a convex subset of a real vector space, then weK defineK ∪ a {{ }} ♯ function κ :Ω d to be concave if for all x,y Ω and all λ (0, 1), we have → K ∈ ∈ λκ(x) + (1 λ)κ(y) κ(λx + (1 λ)y). − ⊂ − If Ω has an ordering, then we define κ to be nondecreasing if for all x,y Ω with x y, we have κ(x) κ(y). ∈ ≤ ⊂ Lemma 27. If κ : [0, ) ♯ is concave, nondecreasing and ∞ → Kd κ(t)= Rd, then the function g : Rd [0, ) defined by t∈[0,∞) → ∞ (60)S g(x)=inf t 0 : x κ(t) { ≥ ∈ } is convex. Furthermore, κ is continuous on (0, ) with respect to the Haus- dorff distance, and for all t> 0, ∞ (61) κ(t)= x Rd : g(x) t . { ∈ ≤ } LAW OF LARGE NUMBERS 27 Proof. By hypothesis, κ(0) = ∅. If 0 / κ(0), then we define κ♯(t)= 6 ♯ ∈ κ(t) x0, where x0 κ(0). The function κ enjoys all of the properties that κ does,− and the function∈ g♯(x)=inf t 0 : x κ♯(t) { ≥ ∈ } ♯ ♯ is related to g by the equation g (x)= g(x + x0). Note that 0 κ (0). If the lemma holds for the functions κ♯ and g♯, it will necessarily hold∈ for κ and g. We may therefore, without loss of generality, assume that 0 κ(0). For any 0 <ε Note that the function g is a generalization of the Minkowski functional of a convex body K, in which case κ(t)= tK. The converse of the preceding lemma also holds; if we are given a convex function g and define κ via (61), then κ is concave. ∞ If (Kn)n=1 is a sequence of convex bodies such that the partial Minkowski N sums SN = n=1 Kn converge in the Hausdorff distance to a convex body ∞ S, then we writeP S = n=1 Kn and refer to this as a Minkowski series. Note that S can also be definedP by the equation ∞ x ◦ = x ◦ . k kS k kKn nX=1 Basic properties of a Minkowski series are easy to prove, and we leave such an investigation to the reader. The following lemma is also fairly straight- forward. 28 D. FRESEN Lemma 28. For each n N, let αn : [0, ) [0, ) be a concave func- tion, and let K be a convex∈ body with 0 K∞ .→ Provided∞ that n ∈ n ∞ α (t) diam(K ) < n n ∞ Xn=1 for all t 0, then the function κ : [0, ) ♯ defined by ≥ ∞ → Kd ∞ κ(t)= αn(t)Kn nX=1 is concave. The space d is separable with respect to dBM, and we shall use a sequence ∞ K (Kn)n=1 that is dense in d. Since dBM is blind to affine transformations, K d we can assume that the John ellipsoid of each Kn is B2 . As coefficients, we shall use the functions 2 2 2−n t, 0 t 22n , αn(t)= 2 ≤2 ≤ 2n , 22n 2 2n −n+2 ≤ −n+2 2n2 = 2 αn(2 ). Hence, 2 d (κ(22n ), K ) 1 + 2−n+2d. BM n ≤ ∞ Thus the sequence (κ(n))n=1 is dense in d. Since each coefficient αn is nondecreasing and concave, κ is concave andK the function g as defined by (60) is convex. Clearly, lim g(x)= . For some c> 0, the function x→∞ ∞ f(x) = 2−g(cx) is the density of a log-concave probability measure µ on Rd. For each n N, Rd −n Rd −1 ∈ D2−n = x : f(x) 2 = x : g(cx) n = c κ(n). Hence the { ∈ ∞ ≥ } { ∈ ≤ } ∞ sequence (D1/n)n=2 is dense in d. By (7), the sequence (F1/n)n=3 is also dense in . K Kd LAW OF LARGE NUMBERS 29 We now use Theorem 1 with q = 1. Let denote a countably dense Kd subset of d, and let K d. For any ε> 0, there exists an increasing K ∈ K ∞ e sequence of natural numbers (kn)1 such that limn→∞ dBM(F1/kn , K) = 1 ∞ d+3 −1 e and n=1 3 (log kn) < ε. By (5), P lim dBM(Pk , K) = 1 n→∞ n with probability at least 1 ε. Since this holds for all ε> 0, K clBM Pn : n N, n d + 1 almost− surely, where cl denotes closure∈ in {with ∈ ≥ } BM Kd respect to dBM. Since this holds for all K d and d is countable, d cl P : n N, n d + 1 almost surely.∈ TheK resultK now follows sinceK ⊂ BM n e e e d is dense{ in ∈ . ≥ } K Kd e Acknowledgments. 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