Intersection bodies, positive definite distributions, and the Busemann-Petty problem

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1 Intersection bodies, positive definite distributions, and the Busemann-Petty problem Alexander Koldobsky American Journal of Mathematics, Volume, Number 4, August 998, pp (Article) Published by The Johns Hopkins University Press DOI:.353/ajm For additional information about this article Access provided by Penn State Univ Libraries (4 Apr 4 9:56 GMT)

2 INTESECTION BODIES, POSITIVE DEFINITE DISTIBUTIONS, AND THE BUSEMANN-PETTY POBLEM By ALEXANDE KOLDOBSKY Abstract. The 956 Busemann-Petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume. In 988, Lutwak introduced the concept of an intersection body which is closely related to the Busemann-Petty problem. We prove that an origin-symmetric star body K in n is an intersection body if and only if kxk, K is a positive definite distribution on n,wherekxk K minfa > : x akg. We use this result to show that for every dimension n there exist polytopes in n which are intersection bodies (for example, the cross-polytope), the unit ball of every subspace of L p, < p is an intersection body, the unit ball of the space `nq, < q <, is not an intersection body if n 5. Using Lutwak s connection with the Busemann-Petty problem, we present new counterexamples to the problem for n 5, and confirm the conjecture of Meyer that the answer to the problem is affirmative if the smaller body is a polar projection body.. Introduction. In 956, Busemann and Petty [3] posed the following problem. Suppose that K and L are origin-symmetric convex bodies in n so that vol n, (K \? ) vol n, (L \? ) for every from the unit sphere S n, in n,where? fx n : (x, ) g is the hyperplane perpendicular to, and vol n, is the (n, )-dimensional volume. Does it follow that vol n (K) vol n (L)? In 988, Lutwak [6] introduced the concept of intersection body closely related to the Busemann-Petty problem. Let L be an origin-symmetric star body in n. As usual, we denote by kxk L minfa > : x alg the norming functional on n generated by L. Throughout the paper, we assume that the norming functional of a star body is a continuous positive function on S n,. A body K in n is called the intersection body of the star body L if the radial function K (x) kxk, K of K at every point Sn, is equal to the (n, )-dimensional volume of the section of L by the hyperplane?. This can be written as follows: for every S n,, kk, K vol n,(l \? ) kk,n+ n, S n, \? L d n, kk,n+ L (), Manuscript received April 3, 997. esearch supported in part by NSF Grant DMS American Journal of Mathematics (998),

3 88 ALEXANDE KOLDOBSKY where 7! f () S n, \ f () d,? S n, is the spherical adon transform defined for every continuous function f on S n,. A body K in n is called an intersection body if there exists a finite Borel (nonnegative) measure on S n, so that kk K,, as functionals on C(Sn, ), where is the finite Borel measure on S n, defined by h, f i h, f i f () d() S n, for every f C(S n, ). Clearly, every intersection body of a star body is an intersection body for which the corresponding measure has continuous density on S n,. The results of Lutwak [6], slightly improved by Gardner [5] and hang [, ], imply that if K is an intersection body then the Busemann-Petty problem has positive answer for every body L (we use the same notation as in the formulation of the Busemann-Petty problem). On the other hand, if L is not an intersection body and has C boundary and positive curvature then there exists a body K so that K and L give a counterexample to the Busemann-Petty problem. A negative answer to the Busemann-Petty problem for n 5 was established in a sequence of papers by Larman and ogers [4] (for n ), Ball [] (n ), Giannopoulos [9] and Bourgain [] (n 7), Gardner [4] and Papadimitrakis [8] (n 5). Gardner [5] gave a positive answer to the problem for n 3by showing that every origin-symmetric convex body in 3 is an intersection body. hang [] claimed that no cube in n with n 4 is an intersection body of a star body and argued that this implies a negative answer to the problem for n 4. In a later paper hang [] used arguments similar to those from [] to claim that no polytope in n, n 4 is an intersection body. Besides, [] includes a counterexample to a conjecture of Meyer [7] that the answer to the Busemann-Petty problem is positive if K is a polar projection body (unit ball of a subspace of L ). Our results disprove the statements from [] mentioned above. Firstly, we show that there exist polytopes in n, n 4 which are intersection bodies. In Section 3, we present a simple calculation showing that the cross-polytope (unitballof `n ) is an intersection body for every n N, and give a precise expression for the corresponding measure. In Section 4 we show that the answer to Meyer s question is positive and, moreover, the unit ball of every subspace of L p, < p is an intersection body. Note that Meyer [7] has proved that the answer to the Busemann-Petty problem is positive if K is a cross-polytope, and proved the Busemann-Petty inequality up to an absolute constant for polar projection bodies. In Section 5 we prove that the unit ball of the space `nq with < q < is not an intersection body if n 5. In particular, this fact provides new counterexamples to the Busemann-Petty problem for every n 5. All these results are consequences of the following connection between inter-

4 INTESECTION BODIES 89 section bodies and positive definite distributions: a star body K is an intersection body if and only if the function kxk K, is a positive definite distribution in n. This fact, in conjunction with techniques developed earlier for calculating the Fourier transform of norm dependent functions [, 3], allows us to find precise expressions for generating measures (or signed measures) of certain star bodies.. Connection between intersection bodies and positive definite distributions. The main tool of this paper is the Fourier transform of distributions. As usual, we denote by S( n ) the space of rapidly decreasing infinitely differentiable functions (test functions) in n,ands ( n ) is the space of distributions over S( n ). The Fourier transform ˆf of a distribution f S ( n )isdefinedby hˆf, ˆi () n h f, i for every test function. A distribution f is called even homogeneous of degree p if h f (x), (xt)i jtj n+p h f, i for every test function and every t, t 6. The Fourier transform of an even homogeneous distribution of degree p is an even homogeneous distribution of degree,n, p. A distribution f is called positive definite if, for every test function, h f, (, x)i. L. Schwartz s generalization of Bochner s theorem states that a distribution is positive definite if and only if it is the Fourier transform of a tempered measure in n ([8, p. 5]). ecall that a (nonnegative, not necessarily finite) measure is called tempered if ( + kxk ), d(x) < n for some >. Every positive distribution (in the sense that h f, i for every nonnegative test function ) is a tempered measure [8, p. 47]. Therefore, a distribution is positive definite if and only if its Fourier transform is a positive distribution. The well-known connection between the adon transform and the Fourier transform is that, for every S n, and every even integrable function on n, the function t! ˆ(t) is the Fourier transform of the function z! (x,)z (x) dx. In fact, making a change of variables (, x) z we get n ˆ(t) (x)exp(, it(, x)) dx exp (, itz) dz (x) dx. (,x)z For every even distribution f S( n )wehave(ˆf )^ () n f. Therefore, the Fourier transform of the distribution t! ˆ(t) is equal to (,x)z (x) dx. If is an even function from L ( n ) and the function t! ˆ(t) belongs to L ( ), we have () ˆ(t)exp(, itz) dt (x) dx. We will use this formula mostly with z. (,x)z

5 83 ALEXANDE KOLDOBSKY Another simple fact, that we often use, is that the norming functional kxk of every star body is equivalent to the Euclidean norm in the sense that, for every x n, K kxk kxk K kxk for some positive constants K, K. Hence, kxk,p is a locally integrable function on n for every p (, n). The following Fourier transform formula for sections of star bodies was shown in []: for every star body L and every S n, vol n, (L \? ) (n, ) (kxk,n+ L )^(). This implies that if K is the intersection body of a star body L then kuk, K (n,) (kxk,n+ L )^(u) for every u n.wehavekxk,n+ L (n,) () n (kuk, K )^(x). Therefore, a star body K is an intersection body of a star body if and only if kxk, K is a distribution whose Fourier transform is a continuous positive function on n n fg. In Theorem, we will show a more general result. We need the following fact. LEMMA. Let be a tempered measure on n, and suppose that is also an even homogeneous distribution of degree,n +. Then there exists a finite Borel measure on the sphere S n, so that for every even test function () n (, ) (x) d(x) d () S n, (t) dt. Proof. Let us first show that can not have an atom at the origin. In fact, suppose that + a, where (fg),and is the unit mass at the origin. Since is homogeneous of degree,n +, for every nonnegative test function with () > and every t >, we have (, (xt)) t(, )! ast!. On the other hand, (, (xt)) (, (xt)) + a(), so a. For every Borel subset A S n, and interval (a, b] [, ) denote by A (a, b] fx n : x t, t (a, b], Ag, andlet A(a,b] be the indicator of this set. By the definition of a homogeneous distribution, we have (, (xt)) t(, ) for every test function and t >. Since is a locally finite measure, we can approximate the functions A(a,b] by test functions with compact support and use the dominated convergence theorem to show that (A [, k]),, A[,] (xk) k(a [, ]). Now, for every Borel subset A S n, and every a < b we have (A (a, b])(b, a)(a [, ]). Define a measure on S n, by (A) (A [, ]) for every Borel set

6 INTESECTION BODIES 83 A S n,. Clearly, d () S n, A(a,b] (t) dt (b, a) (A). Therefore, we get the equality () with A(a,b] and the result follows since A, a, b are arbitrary. THEOEM. A star body K is an intersection body if and only if kxk K, positive definite distribution. is a Proof. Using () and spherical coordinates in the hyperplane?, for every even test function and every S n, we have (3) (t) dt () n ( ˆ)^(t) dt ˆ(x) dx () n, () n,? S n, \? t n, ˆ(t) dt Suppose that K is an intersection body, and let be the measure on S n, for which kk, K. For every nonnegative test function, using (3) and the fact that kxk K, is a locally integrable function, we get ((kxk, K )^, ) (kxk, K, ˆ) kxk, ˆ(x) n K dx kk, S n, K d t n, ˆ(t) dt kk, K, t n, ˆ(t) dt, t n, ˆ(t) dt, t n, ˆ(t) dt Sn,\?, t n, ˆ(t) dt d?, ˆ(x) dx () n,, (t) dt. Therefore, kxk, K is a positive definite distribution. Conversely, if kxk K, is a positive definite distribution then, by L. Schwartz s version of Bochner s theorem [8, p. 5], the Fourier transform of kxk, K is a tempered measure on n. As the Fourier transform of a homogeneous distribution of degree -, this measure is a homogeneous distribution of degree,n +. By Lemma, there exists a finite Borel measure on S n, so that for every d.

7 83 ALEXANDE KOLDOBSKY test function n (, ) (x) d(x) d () S n, Now, by () and (), for every test function On the other hand, (kxk, K, ) () n ((kxk, K )^, ˆ) () n d () S n, () n, d () S n, () n,, (kxk, K, ) () n,, S n, kk, (t) dt. (, ˆ) () n ˆ(t) dt (x) dx (,x) t n, (t) dt t n, (t) dt. K d t n, (t) dt. Every infinitely differentiable function v on the sphere S n, can be represented in the form t n, (t) dt for some S( n ) (take (x) u(t)v() for every x t n, t >, S n,,whereu S( ) is a nonnegative function supported outside of zero and tn, u(t)dt ). Therefore, the measures kk, K and () n,, considered as functionals on C(S n, ), coincide on infinitely differentiable functions on S n,. Hence, those functionals are equal, and K is an intersection body. 3. The cross-polytope is an intersection body. In this section, we show directly (without using Theorem ) that, for every n, the cross-polytope B fx n : kxk P jx i j g is an intersection body. Let us first note that the function exp(,kxk ) is the Fourier transform of the Q function () nk n, and both functions + k and ˆ are integrable on n. Using () and spherical coordinates in the hyperplane (x, ),weget (4) kxk, exp(, jtjkxk ) dt n, S n, \x? ny (x,) n, k ny t n, +t k k dt +k! d. d

8 INTESECTION BODIES 833 This argument suggests that kxk, is the spherical adon transform of the function h() n, t Q n, n k dt, +t k S n,. This function belongs to L (S n, ), but it is not bounded, so B is an intersection body, but not the intersection body of a star body. In this section we give a formal proof of this fact. Q The function () nk n is the density of the standard -stable + k measure on n (the Cauchy distribution). A classical result of P. Levy [5] is that the image of this measure under the mapping! (x, )kxk from n to is equal to the measure (t) dt on (+t independently of the choice of ) x n, x 6. In fact, fix x n, x 6 and denote by x the measure on given by x (A) f n : (x, )kxk Ag for every Borel set A in. Then, for every k exp (, jkjkxk )c (kx) exp (, i(kx, )) d () n exp (, ikkxk ((x, )kxk )) d () n exp (, ikkxk y) d x (y) b x (kkxk ). We have b x (kkxk )exp(, jkjkxk ) for every k,sod x (t) every x n n fg. We need the following fact. LEMMA. The function (+t ) dt for h() n, ny t n, +t k k dt is integrable on S n,. Proof. The function g() Q n k + k belongs to L ( n ), therefore n g() d d S n, t n, NY +t k k dt <, so the function! t n, Q n k +t k dt belongs to L (S n, ). We have n, h() ny t n, ny +t k k dt + t n, +t k k ny t n, +t k k dt L (S n, ). + dt

9 834 ALEXANDE KOLDOBSKY THEOEM. The cross-polytope is an intersection body. Proof. Let be an arbitrary even test function and (, ). Consider the integral ny (5) n j(x, n )j,+ n +k (x) dx d. k By the Fubini theorem, the integral (5) is equal to! ny n j(x, n )j,+ n + d (x) dx k k n j(x, )j,+ ny n kxk n +k k d! kxk,+ (x) dx. Now we make a change of variables t (x, )kxk and apply the remark before Lemma to see that the latter integral is equal to () jtj,+ +t dt n kxk,+ (x) dx. Both integrals converge absolutely, which justifies the use of the Fubini theorem earlier. The integral over is equal to (6) t,+ +t dt t,+ (, ) dt +t + t,+ dt + jtj,+ +t dt. The second summand in (6) is equal to, the first and the third approach zero as!. Since the function kxk, is locally integrable on n, we can apply the dominated convergence theorem to see that n kxk,+ (x) dx! kxk, n (x) dx, as!. In fact, the functions under the integral in the left-hand side are dominated by an integrable function max (, kxk, )j(x)j. Therefore, the limit of the integral (5), as!, is equal to () n kxk, (x) dx. On the other hand, the integral (5) is equal to n n k ny +k j(x, n )j,+ (x) dx d.

10 INTESECTION BODIES 835 Passing to the spherical coordinates in the outer integral and then making a change of variables (x, ) z, weget! ny n t n,+ S n, +t k k dt j(x, n )j,+ (x) dx d ny! n t n,+ dt jzj,+ (x) dx dz d. S n, +t k k (x,)z By the dominated convergence theorem (using estimates similar to those in Lemma ), for almost every S n,, the integral against dt converges to n, h(), and it is dominated by an integrable (on S n, ) function + t Q n, n k dt. +t k Suppose that the test function has compact support D. The function (; z) (x,)z (x) dx is the adon transform of the test function and has continuous derivatives of all orders on S n,. Similarly to (6), we have (7) jzj,+ (; z) dz + z,+ ((; z), (;)) dz z,+ (;) dz + z,+ (; z) dz. The second summand is equal to (; ). The first summand can be estimated by max S n,, z[,] j z(; z)j, and the third is less than or equal to max S n, j(; z)j dz. It follows that the integral (7) converges to (; ) and is dominated by a constant function, as!. Therefore, the integral (5) converges to ()hh(), (x,) (x) dxi, as!, and, as we showed earlier, the limit of (5) is also equal to () n kxk, (x) dx. Suppose that (x) u(z)v() for every x z n, z, S n,,where u is a nonnegative test function on with compact support outside of zero, and v is any infinitely differentiable function on S n,. Using spherical coordinates in the hyperplane (x, ),weget (x) dx z n, u(z) dz v() d. (x,) S n, \(x,) On the other hand, using spherical coordinates in n,weget kxk, n (x) dx z n, u(z) dz kk, S n, v() d.

11 836 ALEXANDE KOLDOBSKY It follows that hh, vi hkk,, vi for every infinitely differentiable function v on S n,. We conclude that kk, h. 4. An affirmative answer to a question of Meyer. Meyer [7] proved that the answer to the Busemann-Petty problem is positive if K is the cross-polytope and L is any origin-symmetric convex body. Now we can also see this fact as a consequence of Theorem and hang s version of Lutwak s connection between intersection bodies and the Busemann-Petty problem mentioned in the Introduction [, Theorem 5]. Meyer [7] also proved the Busemann-Petty inequality up to a constant, not depending on the dimension, for unit balls of subspaces of L q, < q and asked whether the constant can be removed. Our next result gives an affirmative answer to Meyer s question. THEOEM 3. The unit ball of any n-dimensional subspace of L q with < q is an intersection body. Proof. By a well-known result of P. Levy [5], for every n-dimensional subspace B ( n, k k) ofl q with < q, the function exp (, kxk q )isthe Fourier transform of a q-stable symmetric measure on n. We have kxk, q Γ(q) exp (, t q kxk q ) dt. The function kxk, is locally integrable in n and bounded at infinity, hence, for any test function the function kxk, ˆ(x) is integrable on n. Therefore, we can use the Fubini theorem in the following calculation: (8) ((kxk, )^, ) kxk, ˆ(x) dx n q n ˆ(x) dx exp (, t q kxk q ) dt Γ(q) q dt ˆ(x)exp(, t q kxk q ) dx Γ(q) n q dt Γ(q) t,n ˆ(yt)exp(, kyk q ) dy. n The function y! t,n ˆ(yt) is the Fourier transform of the function! (t). Therefore, t,n ˆ(yt)exp(, kyk q ) dy (exp(, kyk q ), t,n ˆ(yt)) n () n (, (t)) () n (tx) d(x). n

12 INTESECTION BODIES 837 Now (8) can be written as ((kxk, )^, ) ()n q Γ(q) dt (tx) d(x). n The latter equality shows that ((kxk, )^, ) if is any nonnegative test function, so (kxk, )^ is a positive distribution, and the result follows from Theorem. It follows from Theorem 3 and [, Theorem 5] that THEOEM 4. If K is the unit ball of any finite dimensional subspace of L q, < q then the answer to the Busemann-Petty problem is positive for every originsymmetric convex body L. 5. Unit balls of the spaces `nq, q >. Let kxk q (jx j q + :::+ jx n j q ) q be the norm of the space `nq, < q <. Denote by q the Fourier transform of the function z! exp (, jzj q ), z. The properties of the functions q were studied by Polya [9]. In particular, if q is not an even integer, the function q (t) behaves at infinity like jtj,q,. Namely (see [, Part 3, Problem 54]), lim t! t+q q (t) Γ(q +)sin(q). If q is an even integer, the function q decreases exponentially at infinity. The integral S q () jtj q (t) dt converges absolutely for every (,, q). These moments can easily be calculated (see [3] or []; is not an even integer): S q () + Γ(, q)γ(( +))(qγ(, )). Clearly, the moment S q () is positive if (,, ) [ (, ), and the moment is negative if (, min (q, 4)). The Fourier transform of the function kxk q was calculated in []. LEMMA 3. Letq >,n N,,n < < qn, q 6 N [fg, (, :::, n ) n, k 6, k n. Then (kxk q )^() q Γ(, q) ny t n+, k q (t k ) dt.

13 838 ALEXANDE KOLDOBSKY Proof. Assume that, < <. By the definition of the Γ function (jx j q + + jx n j q ) q q Γ(, q) y,, exp (, y q (jx j q + + jx n j q )) dy. For every fixed y >, the Fourier transform of the function x! exp (,y q (jx j q + + jx n j q )) at any point n is equal to y,n Q n k q ( k y). Making the change of variables t y we get (9) ((jx j q + + jx n j q ) q )^() q Γ(, q) q Γ(, q) ny y,n,, k ny t n+, k q ( k y) dy q (t k ) dt. The latter integral converges if,n < < qn since the function t! Q nk q (t k ) decreases at infinity like t,n,nq (recall that k 6, k n.) If is allowed to assume complex values then both sides of (9) are analytic functions of in the domain f,n < e < nq, q 6 N [ fgg. These two functions admit unique analytic continuation from the interval (,, ). Thus the equality (9) remains valid for all (,n, qn), q 6 N [fg (see [7] for details of analytic continuation in such situations). Let us prove that the function (kxk,p q )^ is sign-changing for every < p < n,3. The following argument is similar to that used in the proof of Schoenberg s conjecture on positive definite functions in []. LEMMA 4. Ifq >, n > 3, p (, n, 3) then the distribution kxk q,p positive definite. is not Proof. By Lemma 3 and properties of the moments S q (), the integral I(, :::, n, ) j j :::j n, j n, (kxk,p q )^(, :::, n,,) d :::d n, S q ( ) :::S q ( n, )S q (,, :::, n,, p) converges absolutely if the numbers, :::, n,,,,:::, n,,p belong to the interval (,, q). Choosing k (,, ) for every k,:::, n,, we have the moments S q ( k ), k,:::, n, positive, and we can make,,:::, n,,p equal to any number from (, p, n,, p) \ (,, q). Since < p < n, 3, this interval contains a neighborhood of, and, since the moment function S q changes its sign at, we can make the integral I(, :::, n, ) positive for one choice of s and negative for another choice. This means that the function (kxk,p q )^ is sign-changing.

14 INTESECTION BODIES 839 THEOEM 5. The unit ball of the space `nq, < q <, n 5 is not an intersection body. Proof. Forn 5 the number belongs to the interval (, n, 3), so, by Lemma 4, the function kxk q, is not a positive definite distribution. Now the result follows from Theorem. The case of the unit balls of the spaces `4q, < q requires very complicated calculations (see [3]) and will be considered in a forthcoming paper. DIVISION OF MATHEMATICS AND STATISTICS, UNIVESITY OF TEXAS AT SAN ANTONIO, SAN ANTONIO, TX 7849 Electronic mail: KOLDOBSK@MATH.UTSA.EDU EFEENCES [] K. Ball, Some remarks on the geometry of convex sets, Geometric Aspects of Functional Analysis, Lecture Notes in Math, vol. 37, 988, Springer-Verlag, New York, pp [] J. Bourgain, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. (99), 3. [3] H. Busemann and C. Petty, Problems on convex bodies, Math. Scand. 4 (956), [4]. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 34 (994), [5], A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. 4 (994), [6], Geometric Tomography, Cambridge Univ. Press, Cambridge, 995. [7] I. M. GelfandandG. E. Shilov, Generalized Functions. Properties and Operations, Academic Press, New York, 964. [8] I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions 4. Applications of Harmonic Analysis, Academic Press, New York, 964. [9] A. Giannopoulos, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (99), [] A. Koldobsky, Schoenberg s problem on positive definite functions, Algebra i Analiz 3 (99), (ussian); English transl., St. Petersburg Math. J. 3 (99), [], Characterization of measures by potentials, J. Theoret. Probab. 7 (994), [], An application of the Fourier transform to sections of star bodies, Israel J. Math. (to appear). [3], Positive definite distributions and subspaces of L,p with applications to stable processes, Canad. Math. Bull. (to appear). [4] D. G. Larman and C. A. ogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika (975), [5] P. Levy, Théorie de l addition de variable aléatoires Gauthier-Villars, Paris, 937. [6] E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math 7 (988), 3 6. [7] M. Meyer, On a problem of Busemann and Petty, unpublished manuscript. [8] M. Papadimitrakis, On the Busemann-Petty problem about convex, centrally symmetric bodies in n, Mathematika 39 (99), [9] G. Polya, On the zeroes of an integral function represented by Fourier s integral, Messenger Math. 5 (93),

15 84 ALEXANDE KOLDOBSKY [] G. Polya and G. Szego, Aufgaben und lehrsatze aus der analysis, Springer-Verlag, New York, 964. [] G. hang, Intersection bodies and Busemann-Petty inequalities in 4, Ann. of Math. 4 (994), [], Sections of convex bodies, Amer.J.Math. 8 (996), [3] V. M. olotarev, One-dimensional Stable Distributions, Amer. Math. Soc., Providence, 986.