Affine Diameters of Convex Bodies—A Survey

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Expo. Math. 23 (2005) 47–63 www.elsevier.de/exmath

Afﬁne diameters of convex bodies—a survey V. Soltan∗ Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA Received 9 February 2004

Abstract This survey covers various topics related to affinediameters of convex bodies inEuclideanspace. It describes the known properties of affine diameters and contains a list of existing and new open problems. Also, characterizations of simplices and centrally symmetric convex bodies, as well as a descriptionof a measure of symmetry interms of affinediameters, are given.

MSC 2000: primary 52A20; secondary 52A37; 52B11

Keywords: Afﬁne diameter; Convex body; Convex polytope; Critical set; Critical ratio; Measure of symmetry; Simplex; Conjugate diameters

1. Introduction

Although the notionofaffinediameter of a convex body canbe traced inthe literature from the beginning of the 20th century (see, e. g., [7,9,72,73]), it has become a focus of proper interest over the last 50 years. Thus, Hammer [28–30] used affine diameters to characterize centrally symmetric convex bodies. Klee [40], generalizing the results of Neumann [59] and Süss [74], related affine diameters to the critical set of a convex body. Finally, Grünbaum’s survey [27] onmeasures of symmetry of convex bodies widely operates with affine diameters.

∗ Tel.: +1 703 993 1474; fax: +1 703 993 1491. E-mail address: [email protected].

This article is the first survey onvarious topics related to affinediameters of convex bodies inEuclideanspace. It summarizes the knownresults onaffinediameters, scattered inthe research papers on convex geometry, and contains a list of known and new open problems. Because of space limitations, we do not cover the results on extended affine diameters (see [29–31,34,35]) and on affine diameters of typical convex bodies (see [4,79]). The article’s content is described by section headings as follows. 1. Introduction 2. Classes of affine diameters 3. Basic properties of affine diameters 4. Central symmetry and affine diameters 5. The critical ratio and the critical set 6. Spreads of affine diameters 7. A characterizationof simplices 8. Conjugate affine diameters In what follows, by a convex body we mean a compact, convex set with nonempty interior inthe Euclideanspace Ed . A convex body is called strictly convex if its boundary contains no line segment.A chord of a convex bodyK ⊂ Ed is a line segment [a,b], whose endpoints a and b are distinct and both lie in the boundary of K. As usual, bd K and int K stand for the boundary and the interior of K. A convex d-polytope is a convex polytope of dimension d d. Finally, Br (a) ={x ∈ E | x − a r} denotes the close spherical r-neighborhood of a point a ∈ Ed .

2. Classes of afﬁne diameters

From the variety of afﬁne diameters of convex bodies, metric diameters are the most known and studied. Let us recall that a chord [a,b] of a convex body K ⊂ Ed is a metric diameter provided it is a chord of maximum length in K. The following elementary fact on metric diameters is well-knowninconvex geometry.

d 2.1. Let [a,b] be a metric diameter of a convex body K ⊂ E and Ha, Hb be the hyperplanes through a and b, respectively, both orthogonal to [a,b]. Then Ha and Hb support K such that K ∩ Ha ={a} and K ∩ Hb ={b}.

Proof. Assume for a moment that at least one of these hyperplanes, say Ha, either intersects int K or supports K at more than one point. In either case, there exists a point x ∈ K ∩ Ha distinct from a. Since Ha is orthogonal to [a,b],wehave x − b > a − b , contradicting the choice of [a,b]. Hence K ∩ Ha ={a}. By a similar argument, K ∩ Hb ={b}.

A more general notion is that of a double normal: a chord [a,b] of a convex body K ⊂ Ed is called a double normal provided the hyperplanes through a and b, both orthogonal to [a,b], support K. Clearly, any metric diameter of K is a double normal. The existence of a double normal distinct from a given metric diameter follows from the statement below (see, e. g., [9]). V. Soltan/ Expo. Math. 23 (2005) 47–63 49

d 2.2. Let H1,H2 be a pair of distinct parallel hyperplanes in E both supporting a given d convex body K ⊂ E such that the distance between H1 and H2 equals the width of K, i.e., this distance is minimum possible among all such pairs of supporting hyperplanes. Then there are points u ∈ K ∩ H1 and v ∈ K ∩ H2 such that [u, v] is orthogonal to both H1 and H2; whence [u, v] is a double normal of K.

Proof. For any direction l in Ed , denote by s(l) a chord of K that has the maximum length inthe direction l. Let l∗ be a direction that minimizes the length of s(l) over all directions l in Ed , and put [u, v]=s(l∗). As proved in3.1 below, there is a pair of distinct,parallel hyperplanes H1 and H2 both supporting K such that u ∈ K ∩ H1 and v ∈ K ∩ H2.If[u, v] were not orthogonal to both H1 and H2, thenthe direction l orthogonal to H1 and H2 would give a chord s(l ) that is shorter than s(l∗). Hence [u, v] is a double normal of K.

It is interesting to compare the minimum number of metric diameters and that of double normals of a convex body. For example, the ellipsoid x2 x2 x2 d 1 2 d C = (x ,...,xd ) ∈ R + +···+ 1,b>b > ···>bd > 0 1 b2 b2 b2 1 2 1 2 d has one metric diameter (the longest axis of C) and exactly d double normals (the axes of C). Moreover, the following profound statement holds.

2.3. Any convex body in Ed has at least d double normals.

The priority of 2.3 is not clear. It was posed as a problem by Klee [41] in1960 and afﬁrmatively answered by Kuiper [43] in 1964. At the same time, 2.3 is mentioned as a knownfact inLyusternik’s book [49], published in1956, without anyreferenceor proof.

2.4. Deﬁnition. A chord [a,b] of a convex body K ⊂ Ed is said to be an afﬁne diameter of K provided there are two parallel, distinct hyperplanes Ha and Hb both supporting K such that a ∈ Ha and b ∈ Hb.

Unlike double normals, the hyperplanes Ha and Hb from 2.4 are not assumed to be orthogonal to [a,b]. Obviously, affine diameters can be characterized as affine images of double normals. A restricted class of affine diameters is defined by Soltan and Nguyên [69]: a chord [a,b] of a convex bodyK is called an exposed diameter of K provided there are two distinct parallel hyperplanes Ha and Hb both supporting K such that K ∩ Ha ={a} and K ∩ Hb ={b}. If [a,b] is a metric diameter of a convex body K ⊂ Ed , thenfor anyregular affine transformation f : Ed → Ed , the chord [f (a), f (b)] is anexposed diameter of the convex body f(K). Makeev [51] mentioned that the converse statement is not, generally, true. He also observed that if a convex body K ⊂ Ed is smooth and strictly convex, and the Gaussian curvature of bd K is positive, thenanyaffinediameter of K is anaffineimage of a metric diameter. It is easy to see that any convex body K in Ed has infinitely many affine diameters. As proved by SoltanandNguyên [69], a convex body K ⊂ Ed has finitely many exposed 50 V. Soltan/ Expo. Math. 23 (2005) 47–63 diameters if and only if K is a convex polytope; in this case, the number of exposed diameters is at least d, with exactly d if and only if K is a d-cross-polytope. Answering a problem of Blaschke [7], Süss [72,73] and Matsumura [57] (see also [9,12,45]) proved that any planar convex body K has at least three affine diameters with the same curvature of bd K at the end points of each of the diameters.

3. Basic properties of afﬁne diameters

Hammer [30] defined an affine diameter of a convex body K ⊂ Ed as a longest chord of K ina givendirection.Later Hammer [31] made anobservation(without proof) that this definition is equivalent to 2.4. Below we give a proof of this equivalence.

3.1. A chord [a,b] of a convex body K ⊂ Ed is an afﬁne diameter of K if and only if it is a longest chord of K in a given direction.

Proof. Let [a,b] be anafﬁnediameter of K,andHa, Hb be parallel hyperplanes both supporting K such that a ∈ Ha and b ∈ Hb. Denote by l the directionof [a,b].If[x,z] is a chord of K parallel to l, thenfrom the fact that [x,z] lies between Ha and Hb we easily conclude the inequality x − z a − b . Hence [a,b] is a longest chord of K inthe direction l. Conversely, let [a,b] be a longest chord of K ina givendirection l. Consider the convex body K = (a − b) + K. Clearly, a ∈ K . We claim that K and K do not overlap. Indeed, assume, for contradiction, that the bodies have a common interior point, c. Then the point e = c + (b − a) belongs to the interior of K and c − e = a − b . Inthis case, the chord of K through c and e has the direction l and is longer than [a,b], contradicting the choice of [a,b]. Hence K and K have no common interior point, which implies the existence of a hyperplane H separating K and K . Since both K and K contain a, H also contains a. As a result, the hyperplane (b − a)+ H, containing b, also supports K. Thus [a,b] is anafﬁnediameter of K.

By a compactness argument, we conclude from 3.1 the following property of afﬁne diameters (see, e. g., [53]).

3.2. For any direction l in Ed , there is an afﬁne diameter of a convex body K ⊂ Ed in the direction l.

It is interesting that 3.2 was used by Bang [3] in1951 as a knownfact, without any proof or reference. Another basic property of afﬁne diameters, formulated in 3.3 below, was proved by Hammer (see [29] for d = 2and[31] for all d 2).

3.3. Any point of a convex body K ⊂ Ed belongs to an afﬁne diameter of K. V. Soltan/ Expo. Math. 23 (2005) 47–63 51

Proof. Let x be a point in K.Ifx ∈ bd K, thenthere is a hyperplane H through x supporting K.IfH is another hyperplane parallel to H and supporting K, thenfor anypoint y ∈ H ∩K, the chord [x,y] is anafﬁnediameter of K. Now assume that x ∈ int K, and let r be the maximum positive real number with the property (1 + r)x − rK ⊂ K. Due to the choice of r, there exists at least one common boundary point, say a, of the bodies K and (1 + r)x − rK. Denote by H a hyperplane supporting K at a. Clearly, b = (1 + r)x − ra is a boundary point of K such that x ∈[a,b], and the hyperplane H = (b − a) + H contains b and supports K. Inother words, [a,b] is anafﬁnediameter of K.

A study of similar properties of metric diameters and double normals gives the following statement (see [11], [22]).

3.4. For a convex body K ⊂ Ed , the following conditions are equivalent:

(a) for any direction l in Ed , there is a double normal of K parallel to l, (b) any point x ∈ K lies on a double normal of K, (c) every double normal of K is a metric diameter, (d) K is a body of constant width.

4. Central symmetry and afﬁne diameters

If a convex body K ⊂ Ed is symmetric about a point x, thenanychord through x is an afﬁne diameter of K. In1954, Hammer [30] proved the converse statement (a year later independently proved by Buseman [10, pp. 89–90].

4.1. If x is an interior point of a convex body K ⊂ Ed such that every chord through x is an afﬁne diameter of K, then K is symmetric about x.

Proof. Let L be a two-dimensional plane through x,andB be the intersection of K and L. Then B is a planar convex body, and every chord of B through x is anaffinediameter of B. We claim that x bisects every affine diameter of B through it. Let B1 be the reflectionof B B about x. Denote by and 1 the respective boundaries of B and 1. Since x is interior to B, and 1 intersect in at least two points. = () () = Let represent ina polar coordinatesystem with the pole at x. Then 1 ( + ) represents 1. Moreover, since any line through x contains an affine diameter, say [u, v],ofB, there is a unique tangent line l(u) at u, except possibly a countable number of points of , such that the line through v and parallel to l(u) is also a tangent to . Hence / = / = c 1 1 at all except possibly a countable number of points. Hence 1 because c = B = B and 1 are continuous, and 1 because and 1 intersect. As a result, 1 and x is a center of symmetry B. Since L was chosenarbitrarily, K is symmetric about x.

The requirement that x is an interior point is needed in 4.1 since every proper chord of a convex bounded cone through its apex is an afﬁne diameter of the cone. The 52 V. Soltan/ Expo. Math. 23 (2005) 47–63 following simple result gives a similar criterion for strictly convex, centrally symmetric convex bodies.

4.2. For a convex body K ⊂ Ed the following conditions are equivalent:

(a) K is strictly convex and centrally symmetric, (b) there is a point x ∈ int K such that every afﬁne diameter of K contains x.

One more characteristic property of centrally symmetric convex bodies in terms of afﬁne diameters is proved by Falkoner [25] for d = 2 and Khassa [39] for d 3.

4.3. Let K ⊂ Ed be a convex body such that for every direction l and for every pair of parallel hyperplanes H1,H2 both supporting K and parallel to l, there is an afﬁne diameter of K in the direction l lying mid-way between H1 and H2. Then K is centrally symmetric.

4.4. (Harazišvili [36]) If any chord of a planar convex body K that divides the area of K into two parts of equal areas is an afﬁne diameter of K, then K is centrally symmetric.

5. Critical ratio and critical set

Affine diameters are used in the study of critical ratios, also called measures of symmetry, and critical sets of a convex body. The first results on critical ratios were obtained by d Minkowski [58]. Namely, for a givenpoint x of a convex body K ⊂ E , let fH (x) be the ratio, not exceeding 1, in which a hyperplane H through x divides the distance between two supporting hyperplanes of K both parallel to H. Put f(x)=min fH (x), where the minimum is takenover all hyperplanes H through x, and let fK = max{f(x) | x ∈ K}. Minkowski [58], for d = 2, 3 (see also [24]), and later Radon [61], for all d 2, showed that if z is the 1 1 center of gravity of K, then f(z) d , and whence fK d . A significant part of the study on critical ratios is developed by means of chords (see [27]). Let x be an interior point of a convex body K ⊂ Ed . For every chord p =[a,b] through x, denote by rp(x) the ratio, not exceeding 1, in which x divides p: a − x b − x r (x) = , p min x − b x − a .

Obviously, rp(x) is a continuous function in p, with 0

r(x) = max {r 0 | (1 + r)x − rK ⊂ K}.

From the proof of 3.3, we conclude the following observation.

5.1. Let x be an interior point of a convex body K ⊂ Ed . Then any chord of K divided by x in the ratio r(x) is an afﬁne diameter of K. V. Soltan/ Expo. Math. 23 (2005) 47–63 53

The number

rK = max{r(x) | x ∈ K} is called the critical ratio of K (also called Minkowski’s measure of symmetry of K), and the set ∗ K ={x ∈ int K | r(x) = rK } is called the critical set of K. Obviously, K is centrally symmetric if and only if rK = 1; inthis case K∗ consists of a unique point, the center of symmetry of K. As mentioned by d Grünbaum [27], rK = fK for any convex body K ⊂ E . The critical ratio of a planar convex body was initially considered by Neumann [59], who defined the number p(x) for a directed chord p =[a,b] through x as the ratio a − x (x) = p a − b and (x) as the minimum of p(x) over all directed chords p though x. Inthis way, (x) is 1 always at most 2 . This definition was further used in [74,75]. Hammer [28] defined the number rp(x) for a cord p =[a,b] through x as a − x x − b r (x) = , p max a − b a − b and r(x) as the maximum of rp(x) over all chords p though x. As a result, r(x) is always at 1 least 2 . This approach was followed in [6,40,46,78]. Grünbaum’s [27] definition above is used in [16–19,66]. It is compatible with a more general approach of Grünbaum [27] to measures of symmetry of convex bodies in Ed . Clearly, the directionof a chord p divided by x ina critical ratio anda critical set K∗ of the convex body K are the same for any of the above approaches. The lower bound for rK (interms of (x)) was initially studied by Neumann [59], who 1 proved that the critical ratio of a planar convex body K is greater thanor equal to 2 , with r = 1 K 2 if and only if K is a triangle. Süss [74] and independently Hammer [28] showed that d 1 1 if z is the center of gravity of a convex body K ⊂ E , then r(z) d , and, as a result, rK d (see also [6,23,46]). Danzer et al. [13] used Helly’s theorem to show that any convex body d 1 K ⊂ E contains an iterior point x such that rK (x) d . Klee [40] (also [6,46,78]) proved 1 d that the equality rK = d is characteristic for the d-simplexes in E . Negatively answering a question of Hammer [32], Besicovitch and Zamfirescu [5], proved the existence of a planar convex body K and a point x ∈ int K such that the set of all ratios into which x divides the affine diameters through x is uncountable. Neumann [59] showed that the critical set of a planar convex body consists of a single point (see also [18,74]), while A. Sobczyk [29,33] observed that critical points are not generally unique when d 3. For example, the critical set of the prism 3 P ={(x1,x2,x3) ∈ R | 0x1,x2,x3 12,x1 + x2 12}, is the line segment with the end points (4, 4, 3) and (4, 4, 9). 54 V. Soltan/ Expo. Math. 23 (2005) 47–63

The following result, published in [16], is attributed to Hammer (see [40]).

5.2. For any convex body K ⊂ Ed , the critical set K∗ is a closed, convex set.

Proof. Let x and y be any points in K∗. Thenboth sets

Kx = (1 + rK )x − rK K, Ky = (1 + rK )y − rK K are contained in K.Ifz = (1 − )x + y,01, is a point on the segment [x,y], then

(1 + rK )z + rK K ⊂ conv(Kx ∪ Ky) ⊂ K. ∗ ∗ From the deﬁnition of rK it follows that z ∈ K . Hence K is a convex set. By a similar argument, K∗ is closed.

Furthermore, Klee [40] proved that for a convex body K ⊂ Ed , the inequality −1 ∗ rK + dim K d holds. From here we easily conclude that K∗ is of dimension at most d − 2. Indeed, if ∗ rK = 1 then K is centrally symmetric and K consists of a single point. If rK > 1 then ∗ −1 dim K d −rK d − 2. ∗ −1 Inaddition,Klee [40] showed that any point x ∈ K belongs to at least 1 + rK chords of K divided by x inthe ratio rK . Now 5.1 immediately implies the following result, proved by Neumann [59] for d = 2 (see also [18,27,77]).

5.3. For a convex body K ⊂ Ed , any point x in a critical set K∗ belongs to at least three afﬁne diameters of K.

Onecanask whether the numberthree in5.3 is sharp. At least, we canremark that there canbe exactly three chords though a critical point x ∈ K each divided by x inthe ratio rK . d Indeed, let e1,...,ed be the standard basis of E and T be anequilateral triangleinthe plane L(e1,e2), centered at 0. Consider the convex body

K = conv(T ∪{e3, −e3,...,ed , −ed }). r = 1 Clearly, 0 is the only critical point of K, K 2 , and the medians of T are the only chords of K divided by 0 inthe ratio 1 :2. For a givenpoint x of a convex body K ⊂ Ed , Dziechińska-Halamoda and Lasoń [18] characterized the inclusion x ∈ K∗ interms of chords of K (for d = 2), while Dziechińska- Halamoda [16] made a similar characterizationinterms of supportingconesof K (for all d 2). In1981, Romanowicz (see [17,19]) posed the problem onwhether anyconvex body in Ed is the critical set of some convex body of an appropriate dimension. This problem was positively solved by Dziechińska-Halamoda [16] for the case of centrally symmetric convex bodies, and by Dziechińska-Halamoda and Szwiec [19] for the case of convex polytopes. Using the technique elaborated in [16], Laget [44] proved that any d-dimensional convex body is the critical set of a convex body in Ed+2. V. Soltan/ Expo. Math. 23 (2005) 47–63 55

6. Spreads of afﬁne diameters

Dol’nikov [15] observed that any afﬁne diameter of a convex body K in Ed is intersected by another afﬁne diameter of K. The following statement slightly improves this assertion.

6.1. The middle point of any afﬁne diameter of a convex body K ⊂ Ed belongs to another afﬁne diameter of K.

Proof. Let [a,b] be anaffinediameter of K and c be the middle point of [a,b].Ifr(c)= 1, then c is the center of symmetry of K and any chord through c is anaffinediameter of K.If r(c)<1, then, by 5.1, there exists an affine diameter [v,w] of K divided by c inthe ratio r(c). Clearly, [v,w] =[a,b].

Let S(Ed ) denote the family of smooth, strictly convex bodies in Ed ,andG(Ed ) denote the family of convex bodies in Ed , each of them having a C2-smooth boundary of positive Gaussiancurvature. Makeev [50,51] proved the following statements:

(a) any affine diameter of a convex body K ∈ S(Ed ) in Ed is intersected by continuum many affine diameters of K, (b) for any affine diameter p of a convex body K ∈ G(Ed ), there exists a pair of affine diameters of K both intersecting p and forming a preassigned angle between them, (c) if d is even, then for any affine diameter p of a convex body K ∈ G(Ed ), there exists an affine diameter of K intersecting p at a preassigned angle, (d) for any two disjoint affine diameters of a convex body K ∈ G(E3), there is an affine diameter intersecting both of them.

Another type of results on spreads of afﬁne diameters is related to the number of afﬁne diameters that containagivenpoint.In1963, Grünbaum [27] posed the following problem.

6.2. Problem. Does any convex body K ⊂ Ed contain a point that belongs to at least d +1 afﬁne diameters?

Inthe same paper, Grünbaumasked whether every convex body K ⊂ Ed has either continuum many points each belonging to more than one afﬁne diameter, or a point belonging to continuum many afﬁne diameters. Dol’nikov [15] (with some incompleteness) and Harazišvili [37] gave a positive answer to this question (see also [51]). SoltanandNguyên [66] proved the following statement.

6.3. Any convex body K ⊂ Ed has either continuum many points each belonging to at least three afﬁne diameters, or a point belonging to continuum many afﬁne diameters.

Proof. First, we prove the following auxiliary statement: if a given point x ∈ int K belongs to ﬁnitely many, say m, chords of K, each divided by x inthe ratio r(x), thenthere is a neighborhood V ⊂ int K of x such that any point y ∈ V belongs to at least m afﬁne diameters of K. 56 V. Soltan/ Expo. Math. 23 (2005) 47–63

Let li =[ai,bi], i = 1,...,m,be the chords of K divided by x inthe ratio r(x). Put

K = (1 + r(x))x − r(x)K.

By the definition of r(x), the body K lies in K and touches the boundary of K at endpoints of the chords l1,...,lm.Ifr(x)= 1, then x would be the center of symmetry of K and every chord through x would be an affine diameter, contradicting the finiteness hypothesis. Hence r(x)< B (a ) a B (a ) ∩ K 1 and there is a i-neighborhood i i of i such that i i bd lies in K and K a { ,..., } intersects bd only at the point i. Let be any number between 0 and min 1 m such that the neighborhoods B(ai), i = 1,...,m,are pairwise disjoint. Now, consider the family of sets

K() = (1 − )x + K , > 1.

By a continuity argument, there is a number i > 1 such that for any ∈]0, i[, the connected component of K()\K containing the point ai() = (1 − )x + ai lies entirely in B(ai). Analogously, there is a number > 0, such that for every point x inthe -neighborhood

B(x), the connected component of (K() − x + x )\K containing the point ai() − x + x lies entirely in B(ai), i = 1,...,m. Applying a continuous homothetic contraction of

K() − x + x , we obtainthat for some i ∈]0, 1[, the set

(1 − i)x + i(K() − x) touches bd K from withinat a point ai ∈ B(ai). As inthe proof of 5.1, we conclude that ai is an end point of an affine diameter through x. Since the neighborhoods B(ai), i = 1,...,m,are pairwise disjoint, we have m distinct affine diameters through x. Finally, let x be a critical point of K, and denote by L the family of chords of K divided d by x inthe critical ratio rK . Clearly, L is closed inthe naturaltopology of E and contains at least three elements (see 5.3). If there are at least three isolated chords in L, then, by the consideration above, we find a continuum of points each of them lying in at least three affine diameters. If L contains at most two isolated chords l,l , then L\{l,l }, being nonempty, is perfect, and hence contains continuum many elements.

Nguyên [60] proved that any convex d-polytope in Ed has a continuum of points each belonging to at least ﬁve afﬁne diameters. We pose here the following problem.

6.4. Problem. Is it true that any convex body in Ed has either continuum many points each belonging to at least 2d −1 afﬁne diameters, or a point belonging to continuum many afﬁne diameters?

For a convex body K ⊂ Ed , denote by Q(K) the set of all points in the interior of K each of them belonging to an even number of affine diameters of K. SoltanandNguyên [68] showed that for any planar convex body K, the set int K\Q is dense in K. Independently, Makeev [52] proved this result for the case of smooth, strictly convex bodies in the plane. A similar statement was also proved (with some inaccuracies) by Nguyên [60] for the case of a convex d-polytope P ⊂ Ed with the property that each point x ∈ int P belongs to finitely many affine diameters of P. Later Martini and Soltan [55] proved that for any convex V. Soltan/ Expo. Math. 23 (2005) 47–63 57 d-polytope P ⊂ Ed with the property above, the set Q(P ) has d-dimensional measure zero. Based onthese results, the followingproblem is posed in [55].

6.5. Problem. Let K ⊂ Ed be a convex body and Q(K) be the set of points in int K each of them belonging to an even number of afﬁne diameters. Is Q(K) of d-dimensional measure zero?

The following statement of Hammer [29] originated the study of extended afﬁne diameters of convex bodies in two and three dimensions (see [29–31,34,35]).

6.6. For a given convex body K ⊂ Ed , any point in Ed belongs to the line extending an afﬁne diameter of K.

Proof. Let x be a point in Ed . Due to 3.3, it suffices to consider the case x ∈ Ed \K. Consider a ray with the apex x that intersects K in boundary points v and w so that [v,x] contains w. Inparticular, v may coincide with w ona supportingray, or [v,w] may be a chord of K onits boundary. The ratio v − w / v − x varies continuously as a function of the angle formed by the ray [x,v) with a fixed direction. Hence the ratio must attain a maximum value, . Let K = x + (1 − )(K − x), i. e., K is the image of K by the similarity about x with coefficient 1−. We claim that K and K have no interior points in common. Indeed, assume for contradiction, the existence of a point z that belongs to both int K and int K . Let l be the ray through z with apex x. Then l intersects K in boundary points v1 and w1 so that [v1,x] contains w1. Similarly, l intersects K at boundary points v = x + ( − )(v − x), w = x + ( − )(w − x) 1 1 1 1 1 1 [v ,x] w v ∈]v ,z[⊂ such that 1 contains 1. Moreover, from our assumptionit follows that 1 1 ]v1,w1[. As a result,

v − w v1 − v 1 1 > 1 = , v1 − x v1 − x contradicting the choice of . Hence K and K have no interior points in common, and there is a hyperplane H through w supporting both K and K . Obviously, the hyperplane through v parallel to H also supports K.

7. A characterization of simplexes

In1951, Hammer [29] asked whether the d-simplex is characterized among convex bodies in Ed ,d2, by the property that any of its interior points lies on precisely d + 1affine diameters. Two years later Eggleston [20] (see also [21]) answered positively Hammer’s questionfor the case d = 2. Eggleston’s result was sharpened by Soltan and Nguyên [68]: a planar convex body is a triangle if and only if any of its interior points belongs to at least two but finitely many affine diameters of K. Infact, the last statementis animmediate consequence of the following result from [68]. 58 V. Soltan/ Expo. Math. 23 (2005) 47–63

7.1. For a planar convex body K, the following conditions are equivalent:

(a) any interior point of K belongs to at least two affine diameters of K, (b) either K is a triangle or any interior point of K belongs to infinitely many affine diameters of K.

One can easily see that number d + 1 inHammer’s questionabove is incorrectfor d 3. Indeed, if S is a 3-simplex in E3, then any interior point of S belongs to exactly seven affine diameters of S: four of them contain, respectively, the vertices of S and another three join the pairs of crossing edges of S. By a similar consideration, one can easily conclude that for any interior point x of a d-simplex S ⊂ Ed and for any pair of disjoint faces F,F of S with the property dim F + dim F = d − 1, there exists anaffinediameter [a,b] through x such that a ∈ rint F and b ∈ rint F . As a result, x belongs to exactly 2d − 1 affine diameters of S. A corrected versionof Hammer’s questionfor the case d 3 was proposed by Soltanand Nguyên [67] as follows.

7.2. Problem. Is it true that the d-simplex is the only convex body in Ed such that any of its interior points belongs to exactly 2d − 1 afﬁne diameters?

A positive answer to this problem was given by Martini et al. [54] for the case of convex d-polytopes. Later their result was sharpened in [65] as follows. Let P is a convex d-polytope with the vertex-set {a1,...,an}. For any real number r>0, put

Pr =∪{P ∩ Br (ai) | 1i n}, where Br (ai) denotes the r-spherical neighborhood of ai.

7.3. For a convex d-polytope P, the following conditions are equivalent:

(a) P is a d-simplex, (b) any point x ∈ int P belongs to exactly 2d − 1 affine diameters of P, (c) for any r>0, there is a countable set Z ⊂ int P dense in Pr and such that any point x ∈ Z belongs at least 2d − 2 but finitely many affine diameters of P.

We pose here the following new problem.

7.4. Problem. Is it true that a convex body K ⊂ Ed is a simplex if and only if any of its interior points belongs to at least 3 · 2d−2 − 1 but ﬁnitely many afﬁne diameters of K?

Problem 7.4 is partly conﬁrmed by the following two statements, proved in [60,65], respectively.A convex 3-polytope P =conv(a1, a2, a3, a4, a5) is called a bisimplex provided it can be represented as the union of two non-overlapping simplices conv(a1, a2, a3, a4) and 3 conv(a2, a3, a4, a5) such that any plane L ⊂ E parallel to a facet F of P and supporting P (F⊂ / L) intersects P only at one of the points a2,a3,a4. V. Soltan/ Expo. Math. 23 (2005) 47–63 59

7.5. A convex 3-polytope P is either a simplex or a bisimplex provided any of its interior points belongs to at least four but ﬁnitely many afﬁne diameters of P.

7.6. Let P be a convex 3-polytope satisfying the condition: for any r>0 there is a countable set Z ⊂ int P dense in Pr such that any point x ∈ Z belongs to at least five but finitely many affine diameters of P. Then P is either a simplex or a bisimplex.

From either of 7.5 or 7.6 we easily derive that a convex 3-polytope in E3 is a simplex if and only if any of its interior points belongs to at least five but finitely many affine diameters.

8. Conjugate afﬁne diameters

This sectiongives a brief overview of results onconjugateaffinediameters of convex bodies. For additional information on this topic, see Gruber [26], Heil and Krautwald [38], and a recent survey of Martini et al. [56]. Two affine diameters of a planar convex body K are called conjugate provided there is a parallelogram P circumscribed about K such that the sides of P are parallel to the diameters. Clearly, the endpoints of these affine diameters belong to the sides of P. Conjugate affine diameters were used as a geometric tool for the study of orthogonality in linear normed spaces. This explains why conjugate affine diameters were first studied for the case of centrally symmetric convex bodies, associated with the unit balls of a linear normed space. It seems that Auerbach [1] was the first who proved that any centrally symmetric convex body K in the plane has a pair of conjugate affine diameters. His proof is based on the fact that the diagonals of an inscribed in K parallelogram of maximum area are conjugate affine diameters of K. Another way of finding conjugate diameters is due to Day [14], who showed that if P is a parallelogram of minimum area circumscribed about a centrally symmetric convex body K in the plane, then the segments joining midpoints of opposite sites of P are conjugate affine diameters of K. Lenz [48] shows that these two ways always give different pairs of conjugate affine diameters, unless K is determinedby a Radoncurve; inthe last case anyaffinediameter of K has a conjugate affine diameter. A Radon curve (see [62]) is defined as a regular affine image of a convex curve C with center 0 in the plane, such that its polar is a 90o rotationabout 0. As it is shownin [62], a centrally symmetric convex body K inthe plane is bounded by a Radon curve if and only if any affine diameter of K has a conjugate affine diameter. A non-symmetric generalization of Radon curves is introduced by Blaschke [8], who defined P-curves as closed convex curves in the plane, which have a continuous family of inscribed convex quadrilaterals of maximum area. Blaschke proved that: (i) P-curves are exactly the convex curves for which any affine diameter has a conjugate affine diameter, (ii) Radoncurves are centrallysymmetric P-curves. Probably, it was Stein [70,71], who first observed the existence of a pair of conjugate affine diameters of any planar convex body. This fact was independently reproved by Sobczyk [64]. Finally, Heil and Krautwald [38], generalizing the considerations above, showed that 60 V. Soltan/ Expo. Math. 23 (2005) 47–63 for a convex body K inthe plane:

(a) the diagonals of an inscribed convex quadrilateral of maximum area are conjugate affine diameters of K, (b) if P is a parallelogram of minimum area circumscribed about K, then K has a pair of conjugate affine diameters parallel to the sides of P, (c) either the pairs of conjugate affine diameters from (a) and (b) are different, or there are infinitely many pairs of conjugate affine diameters of K.

Now we will define conjugate diameters in higher dimensions. If K is a convex body inEd , d 2, thena family of d affine diameters s1,...,sd of K is said to be conjugate provided the d directions of s1,...,sd span E (equivalently, these directions are linearly independent) and the end-points of any diameter si, i = 1,...,d,belong, respectively, to two different hyperplanes Hi,Hi both supporting K and parallel to every diameter s1,...,si−1,si+1,...,sd . It is easy to see that if K is symmetric about a point z, thenthe diameters s1,...,sd canbe chosento have z as a commonpoint. Although, Banach [2] attributed the existence of conjugate affine diameters of a centrally symmetric convex body K ⊂ Ed to Auerbach, the first published proofs of this fact are due to Day [14] and Taylor [76], who showed that if P is a d-parallelotope of minimum volume circumscribed about K, then the segments joining midpoints of opposite facets of P are conjugate affine diameters of K. Taylor [76] (and later [63]) proved that the diagonals of a d-cross-polytope of maximum volume inscribed in K are also conjugate diameters of K. Sobczyk [64] observed that the existence of conjugate affine diameters of an arbitrary convex body K in Ed follows from the centrally symmetric case. Indeed, considering the centrally symmetric convex body M = K − K and choosing conjugate affine diameters s1,...,sd of M, one gets a system of d conjugate affine diameters of K parallel to s1,...,sd , respectively. Clearly, the affine diameters of K do not necessarily have a point in common. Sobczyk [64] conjectured that any convex body K ⊂ Ed has a system of d conjugate affine diameters with a commonpoint. Higher-dimensional analogues of Radon curves and P-curves lead to affine images of balls and bodies of constant width. Namely, a centrally symmetric convex body K ⊂ Ed is an ellipsoid if and only if it satisfies the following property: any affine diameter of K belongs to a system of d conjugate affine diameters of K (see [47]). The same property for anarbitrary convex body K ⊂ Ed takes place if and only if K is anaffineimage of a body of constant width (see [42]).

References

[1] H. Auerbach, Über eine Eigenschaft der Eilinien mit Mittelpunkt, Ann. Soc. Polon. Math. 9 (1930) 204. [2] S. Banach, Théorie des Opérations Linéares, Monograﬁas de Matematica, vol. 1, Subwncji Funduszu Narodowej, Warszawa, 1932; Engl. Transl.: Theory of Linear Operations, North-Holland Mathematical Library, vol. 38, North-Holland, Amsterdam, 1987 (MR 88a:01065). [3] T. Bang, A solution of the “plank problem”, Proc. Amer. Math. Soc. 2 (1951) 990–993 (MR 13,769a). [4] I. Bárány, T. Zamﬁrscu, Diameters in typical convex bodies, Canad. J. Math. 42 (1990) 50–61 (MR 91a:52002). V. Soltan/ Expo. Math. 23 (2005) 47–63 61

[5] A.S. Besicovitch, T. Zamfirescu, Onpencilsof diameters inconvex bodies, Rev. RoumaineMath. Pures Appl. 11 (1966) 637–639 (MR 34#674). [6] B.J. Birch, On3N pointsina plane,Proc. Cambridge Philos. Soc. 55 (1959) 289–293 (MR 22#201). [7] W. Blaschke, Aufgaben540 und541, Arch. Math. Phys. 26 (1917) 65. [8] W. Blaschke, Zur Affingeometrie der Eilinien und Eiflächen, Math. Nachr. 15 (1956) 258–264 (MR 18,922c). [9] T. Bonnesen, W. Fenchel, Theorie der konvexenK örper, J. Springer, Berlin, 1934; Reprints: Chelsea Publishing Co., Bronx, NY, 1971; T. Bonnesen, W. Fenchel, Springer, Berlin, 1974; Engl. Transl.: Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987 (MR 51#8954; 49#9736; 88j:52001). [10] H. Busemann, The geometry of Geodesics, Academic Press, New York, 1955 (MR 17,779a). [11] G.D. Chakerian, H. Groemer, Convex bodies of constant width, in: P.M. Gruber, J.M. Wills (Eds.), Convexity and its Applications, Birkhäuser, Basel, 1983, pp. 49–96 (MR 85f:52001). [12] W. Cie´slak, S. Gó´zd´z, Oncurves which boundspecial convex sets, Serdica 13 (1987) 281–286 (MR 88h:52005). [13] L. Danzer, B. Grünbaum, V. Klee, Helly’s Theorem and its Relatives, Convexity (Seattle, WA, 1961), Proceedingsof the Symposium onPure Mathematicas, vol.VII,AmericanMathematical Society, Providence, RI, 1963, pp. 101–179 (MR 28#524). [14] M.M. Day, Polygons circumscribed about closed convex curves, Transl. Amer. Math. Soc. 62 (1947) 315–319 (MR 9,246h). [15] V.L.Dol’nikov, On a question of Grünbaum, Issled. po teorii funktsii mnogih veschestv. Peremennih, Jaroslav. Gos. University, Yaroslavl’, 1976, pp. 34–35 (in Russian) (MR 58#30743). [16] Z. Dziechcińska-Halamoda, Characterization of the set of centers of a convex body by support cones, Zeszyty Nauk. Wy˙z. Szkoly Ped. w Opolu Mat. Ser. A. Mat. 21 (1979) 93–100 (inPolish) (MR 82i:52009). [17] Z. Dziechcińska-Halamoda, On centers of a Cartesian product of convex sets, Zeszyty Nauk. Wy˙z. Szkoly Ped. w Opolu Mat. Ser. A. Mat. 22 (1981) 7–11 (inPolish) (MR 83j:52003). [18] Z. Dziechcińska-Halamoda, A. Lasoń, Some properties of the centre of a convex set in the plane, Zeszyty Nauk. Wy˙z. Szkoly Ped. w Opolu Mat. Ser. A. Mat. 16 (1973) 53–60 (inPolish) (MR 53#14307). [19] Z. Dziechińska-Halamoda, W. Szwiec, On critical sets of convex polyhedra, Arch. Math. (Basel) 44 (1985) 461–466 (MR 86j:52003). [20] H.G. Eggleston, Some properties of triangles as extremal convex curves, J. London Math. Soc. 28 (1953) 32–36 (MR 14,896d). [21] H.G. Eggleston, Problems in Euclidean Space: Applications of Convexity, Pergamon Press, London, 1959 (MR 23#A3228). [22] H.G. Eggleston, Sets of constant width in finite dimensional Banach spaces, Israel J. Math. 3 (1965) 163–172 (MR 34,583). [23] E. Ehrhart, Sur les ovales et les ovoïdes, C.R. Acad. Sci. Paris 240 (1955) 583–585 (MR 16,740b). [24] Th. Estermann, Über den Vektorenbereich eines konvexenK örpers, Math. Z. 28 (1928) 471–475. [25] K.J. Falconer, Applications of outwardly simple line families to plane convex sets, Math. Proc. Cambridge Philos. Soc. 82 (1977) 353–356 (MR 56#3744). [26] P.M. Gruber, Radons Beiträge zur Konvexität, in: J. Radon, Gesammelte Abhandlungen, Band 1, Verlag der Österreichischen Akademie der Wissenschaften, Vienna; Birkhäuser Verlag, Basel, 1987, pp. 331–342 (MR 89i:01142a). [27] B. Grünbaum, Measures of symmetry for convex sets, Convexity (Seattle, WA, 1961), Proceedings of the Symposium onPure Mathematics, vol. VII, AmericanMathematical Society, Providence,RI, 1963, pp. 233–270 (MR 27#6187). [28] P.C. Hammer, The centroid of a convex body, Proc. Amer. Math. Soc. 2 (1951) 522–525 (MR 14,687d). [29] P.C. Hammer, Convex bodies associated with a convex body, Proc. Amer. Math. Soc. 2 (1951) 781–793 (MR 14,678e). [30] P.C. Hammer, Diameters of convex bodies, Proc. Amer. Math. Soc. 5 (1954) 304–306 (MR 15,819b). [31] P.C. Hammer, Convex curves of constant Minkowski breadth, Convexity (Seattle, WA, 1961), Proceedings of the Symposium onPure Mathematical, vol. VII, AmericanMathematical Society, Providence,RI, 1963, pp. 291–304 (MR27#4136). [32] P.C. Hammer, Problem 14, Proceedings of the Colloquium on Convexity (University of Copenhagen, 1965), KZbenhavens, KZbenhavens Universitets Matematiske Institut, 1967, pp. 316–317. [33] P.C. Hammer, A. Sobczyk, Abstract 112: critical points of a convex body, Bull. Amer. Math. Soc. 57 (1951) 127. [34] P.C. Hammer, A. Sobczyk, Planar line families. I, Proc. Amer. Math. Soc. 4 (1953) 226–233 (MR 14,787a). [35] P.C. Hammer, A. Sobczyk, Planar line families. II, Proc. Amer. Math. Soc. 4 (1953) 341–349 (MR 15,149e). [36] A.B. Harazišvili, Some properties of the center of symmetry, Soobsch. Akad. Nauk. Gruzin SSR 89 (1978) 21–24 (MR 82g:52007). 62 V. Soltan/ Expo. Math. 23 (2005) 47–63

[37] A.B. Harazišvili, Affine diameters of convex bodies, Soobsch. Akad. Nauk. Gruzin SSR 90 (1978) 541–544 (inRussian)(MR 80a:52002). [38] E. Heil, W. Krautwald, Konjugierte durchmesser und extremale vierecke konvexer Bereiche, Elementary Math. 24 (1969) 97–103 (MR 40#3431). [39] D.S. Khassa, Relationbetweenmaximal chords andsymmetry for convex sets, J. LondonMath.Soc. 15 (1977) 541–546 (MR 56#3745). [40] V. Klee, The critical set of a convex body, Amer. J. Math. 75 (1953) 178–188 (MR 14,678f). [41] V. Klee, Unsolved Problems in Intuitive Geometry, Mimeographed Notes, University of Washington, 1960. [42] W. Krautwald, Kennzeichnungen der affiner Bilder von Körpern konstanter Breite, J. Geom. 15 (1980) 140–148 (MR 82j:52001). [43] N.H. Kuiper, Double normals of convex bodies, Israel J. Math. 2 (1964) 71–80 (MR 30#4191). [44] B. Laget, Sur l’ensemble critique d’un ensemble convexe, Arch. Math. (Basel) 49 (1987) 333–334 (MR 88j:52009). [45] D. Laugwitz, Differentialgeometrie, Teubner Verlagsgesellschaft, Stuttgart, 1960, Engl. Transl.: Differential and Riemannian Geometry, Academic Press, New York, 1965 (MR 22#7061; 30#2406). [46] K. Leichtweiss, Zwei extremalprobleme der minkowski-geometrie, Math. Z. 62 (1955) 37–49 (MR 16,1047d). [47] H. Lenz, Eine Kennzeichnung des Ellipsoids, Arch. Math. (Basel) 8 (1957) 209–211 (MR 19,977d). [48] H. Lenz, Die Eilinien mit einer schar konjugierter Durchmesserpaare, Arch. Math. (Basel) 9 (1958) 134–139 (MR 21#2204). [49] L.A. Lyusternik, Convex Figures and Polyhedra, GITTL, Moscow, 1956 (in Russian); Engl. Transl.: Dover, New York, 1963; Heath & Co., Boston, 1966 (MR 19,57h; 28#4427; 36#4435). [50] V.V. Makeev, Spatial generalizations of certain theorems on convex figures, Mat. Zametki 36 (1984) 405–415 (in Russian); Engl. Transl.: Math. Notes 36 (1984) 700–705 (MR 86i:52002). [51] V.V. Makeev, On intersections of affine diameters of a convex body, Ukrain. Geom. Sb. 33 (1990) 70–73 (in Russian); Engl. Transl.: J. Soviet Math. 53 (1991) 511–513 (MR 91k:52006). [52] V.V. Makeev, A kinematic formula for affine diameters and affine medians of a convex figure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 280 (2001) (in Russian); Geom. i Topol. 7 (2001) 234–238 (MR 86i:52002). n [53] J.T. Marti, Konjugierte Stützhyperebene von konvexenK örpernim R , Elem. Math. 31 (1976) 81–88 (MR 53#14311). [54] H. Martini, M.H. Nguên, V. Soltan, On Eggleston’s theorem about affine diameters, Mathematika 37 (1990) 81–84 (MR 91g:52002). [55] H. Martini, V. Soltan, A theorem on affine diameters of convex polytopes, Acta Sci. Math. (Szeged) 69 (2003) 431–440. [56] H. Martini, K. Swanepoel, G. Weiss, The geometry of Minkowski spaces—a survey. I, Expo. Math. 19 (2001) 97–142 (MR 2002h:46015a). [57] A. Matsumura, Relativgeometrische Erweiterung eines Satzes von W. Blaschke, Japanese J. Math. 5 (1928) 265–267. [58] H. Minkowski, Allgemeine lehrsätze über konvexe polyeder, Nachr. Ges. Wiss. Göttingen, Math-Phys. Kl. (1897) 198–219; also in: Gesamelte Abhandlungen, vol. 2, Teubner, Leipzig, 1911, pp. 103–121. [59] B.H. Neumann, On some affine invariants of closed convex regions, J. London Math. Soc. 14 (1939) 262–272 (MR 1,158a). [60] M.H. Nguyên, Affine diameters of convex bodies, Ph.D. Thesis, State University of Moldova, Chi¸sin˘au, 1990 (inRussian). [61] J. Radon, Über eine Erweiterung des begriffs der konvexenFun ctionen, mit einer Anwendung auf die Theorie der konvexen Körper, S.-B. Akad. Wiss. Wien125 (1916) 241–258. [62] J. Radon, Über eine besondere art ebener konvexer kurven, Ber. Verh. Sächs. Akad. Leipzig 68 (1916) 123–128. [63] A.F. Ruston, Auerbach’s theorem and tensor products of Banach spaces, Proc. Cambridge Philos. Soc. 58 (1962) 476–480 (MR 29#2630). [64] A. Sobczyk, Convex polygons, Proc. Amer. Math. Soc. 15 (1964) 438–446 (MR 28#4429). [65] V. Soltan, Some notes on Eggleston’s result about affine diameters, Intuitive Geometry (Szeged, 1991), Colloquie. Mathematica Societatis Janos Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 439–446 (MR 97b:52002). [66] V. Soltan, M.H. Nguyên, On the Grünbaum problem on affine diameters, Soobsch. Akad. Nauk Gruzin. SSR 132 (1988) 33–35 (inRussian)(MR 90k:52005). [67] V. Soltan, M.H. Nguyên, Affine diameters of convex bodies, Geometrie und Anwendungen, Materialen zur 7, Tagung der Fachsektion Geometrie, Mathematische Gesellschaft der DDR, Berlin, 1988, pp. 109–110. [68] V. Soltan, M.H. Nguyên, Some notes about affine diameters of convex figures, Combinatorica 10 (1990) 313–317 (MR 92b:52005). V. Soltan/ Expo. Math. 23 (2005) 47–63 63

[69] V. Soltan, M.H. Nguyên, Lower bounds for the number of extremal and exposed diameters of a convex body, Studia Math. Hungar. 28 (1993) 99–104 (MR 94j:52014). [70] S.K. Stein, A continuous mapping defined by a convex curve, Math. Z. 68 (1957) 282–283 (MR 19,877a). [71] S.K. Stein, A continuous mapping defined by a convex curve (addendum), Math. Z. 70 (1958/1959) 465 (MR 21#1565). [72] W. Süss, Kurzer Beweis eines Satzes von W. Blaschke über Eilinien, Tôhoku Math. J. 24 (1924) 66–67. [73] W. Süss, Einige mit dem vierscheitelsatz für eilinien zusammenhängende Sätze, Tôhoku Math. J. 28 (1927) 216–220. [74] W. Süss, Über eineAffininvariante von Eibereichen,Arch. Math. (Basel) 1 (1948/49) 127–128 (MR 10,395b). [75] W. Süss, Über Eibereiche mit Mittelpunkt, Math. Phys. Semesterber. 1 (1950) 273–287 (MR 12,46d). [76] A.E. Taylor, A geometric theorem and its applications to biorthogonal systems, Bull. Amer. Math. Soc. 53 (1947) 614–616 (MR 8,588c). [77] I.M. Yaglom, V.G. Boltyanski, Convex Figures, GITTL, Moscow, 1951 (in Russian); Engl. transl.: Holt, Rinehart & Winston, New York, 1961 (MR 14,197d; 18,146f; 23#A1283). [78] T. Zamfirescu, Two characterizations of simplexes, Gaz. Mat. Ser. A 71 (1966) 422–424 (in Romanian) (MR 38#1612). [79] T. Zamfirescu, Intersecting diameters in convex bodies, Convexity and Graph Theory (Jerusalem, 1981), pp. 311–316, North-Holland Mathematical Studies, vol. 87, North-Holland, Amsterdam, 1984 (MR 87a:52013).