The complex Busemann-Petty problem on sections of convex bodies

THE COM PLEX BUSEMANN-PETTY PR OBLEM ON SECTIONS OF CONVEX BODIES A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOULOU Abstract. The co mplex Busemann-Pett y pro blem asks whether origin sy mmetric conv ex b o dies in C n with smaller central hyp e r - plane sections necessar ily hav e smaller v olume. W e prov e that the answer is affirmative if n ≤ 3 and negative if n ≥ 4 . 1. Introduction The Busemann-P e tt y pro blem, p osed in 1956 (see [BP]), asks the follo wing question. Supp ose that K and L are o r igin symmetric conv ex b o dies in R n suc h that V ol n − 1 ( K ∩ H ) ≤ V ol n − 1 ( L ∩ H ) for ev ery cen tral hy p erplane H in R n . Do es it follow that V ol n ( K ) ≤ V ol n ( L )? The answ er is affirmative if n ≤ 4 and negativ e if n ≥ 5 . The solution w as comple ted in the end o f the 90’s as the result of a sequence of pap ers [L R], [Ba], [G i], [Bo], [Lu], [P a], [Ga1], [Ga2], [Zh1 ], [K1 ], [K 2 ], [Zh2], [GKS] ; see [K10, p. 3 ] for the history of the solution. In t his article w e consider the complex ve rsion of the problem. F or ξ ∈ C n , | ξ | = 1 , denote b y H ξ = { z ∈ C n : ( z , ξ ) = n X k =1 z k ξ k = 0 } the complex hyperplane p erp endicular to ξ . Origin symmetric con ve x b o dies in C n are the unit balls of norms on C n . W e denote by k · k K the norm corresponding to the b o dy K : K = { z ∈ C n : k z k K ≤ 1 } . In o r der to define volume, w e iden tify C n with R 2 n using the mapping ξ = ( ξ 1 , ..., ξ n ) = ( ξ 11 + iξ 12 , ..., ξ n 1 + iξ n 2 ) 7→ ( ξ 11 , ξ 12 , ..., ξ n 1 , ξ n 2 ) . 1 2 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU Under this mapping the hyperplane H ξ turns into a tw o -co dimensional subspace of R 2 n orthogonal to the ve ctors ξ = ( ξ 11 , ξ 12 , ..., ξ n 1 , ξ n 2 ) and ξ ⊥ = ( − ξ 12 , ξ 11 , ..., − ξ n 2 , ξ n 1 ) . Since norms on C n satisfy the equ alit y k λz k = | λ |k z k , ∀ z ∈ C n , ∀ λ ∈ C , origin symmetric complex con vex b o dies corresp ond to those orig in symmetric con ve x b o dies K in R 2 n that are in v ariant with resp ect to an y co ordinate-wise tw o-dimensional rotation, namely for eac h θ ∈ [0 , 2 π ] and eac h ξ = ( ξ 11 , ξ 12 , ..., ξ n 1 , ξ n 2 ) ∈ R 2 n k ξ k K = k R θ ( ξ 11 , ξ 12 ) , ..., R θ ( ξ n 1 , ξ n 2 ) k K , (1) where R θ stands for the coun terclo c kwise r o tation o f R 2 b y the a ngle θ with resp ect to the origin. W e shall simply sa y that K is i n variant with r esp e ct to a l l R θ if it satisfies the equations (1). No w the complex Busemann-P ett y problem can b e f orm ulated a s follo ws: supp ose K and L are origin symm etric inv ariant with resp ect to all R θ con ve x b o dies in R 2 n suc h that V ol 2 n − 2 ( K ∩ H ξ ) ≤ V ol 2 n − 2 ( L ∩ H ξ ) for eac h ξ from the unit sphere S 2 n − 1 of R 2 n . Do es it follow tha t V ol 2 n ( K ) ≤ V ol 2 n ( L )? This form ulation reminds of the lo we r-dimensional Busemann-P ett y problem, whe re one tries to deduce the inequality for 2 n - dimensional v olumes of arbitrary or ig in-symmetric conv ex b o dies from the inequal- ities for v olumes o f all (2 n − 2)-dimensional sections. In t he case where n = 2 this amoun t s to considering t wo-dimens ional sections of four- dimensional b o dies, where the answe r to the low er dimensional problem is affirmative by the solution to the orig inal Busemann-P etty pro blem - w e first get inequalities for the v olumes o f a ll three-dimensional sections and then the inequalit y for the four- dimensional v olumes. Ho wev er, if n = 3 we get four- dimensional sections of six-dimensional b o dies, where the a nswe r to the lo w er-dimensional problem is negative b y a result of Bourgain and Zhang [BZ]. Our pr o blem is differen t from the lo w er- dimensional Busemann-P ett y problem in t wo asp ects. First, w e do not ha ve all (2 n − 2)-dimensional sections, w e o nly hav e sections by sub- spaces coming from complex h yp erplanes, whic h makes the situation w orse than f or the low er-dimensional problem. Secondly , w e conside r only those conv ex b o dies in R 2 n that ar e inv ariant with respect to all R θ , and w e ma y b e able to conv ert this in v ariance in to affirmative answ er in some higher dimensions. COMPLEX BUSEMANN-PETTY PROBLEM 3 The latter app ears to b e the case, a s we pro v e b elo w that the answ er to the complex Busemann-P etty problem is affirmative if n ≤ 3 and negativ e if n ≥ 4 . In 1988 Lut w ak [Lu] in tr o duced the class of interse ction b o dies and found a connection b et w een this class and the “real” Busemann-P ett y problem, whic h play ed an imp ortan t ro le in the solution of the problem. It app ears that the complex Busemann-P etty problem is closely related to the class of 2-inters ection b o dies in tro duced in [K5, K8], namely the answ er to the problem is affirmativ e if a nd only if ev ery origin symmetric in v aria n t with resp ect to all R θ con ve x b o dy in R 2 n is a 2- in tersection bo dy . W e shall pro v e this connection in The orem 2. After that w e pro ve tha t eve ry origin symmetric in v a r ian t with resp ect to all R θ con ve x b o dy in R 2 n is a (2 n − 4)- in tersection b o dy , but not ev ery suc h b o dy is a (2 n − 6 )-in tersection b o dy . Putting n = 3 and then n = 4, one can see ho w these results imply the solution of the complex Busemann-P etty problem. Our pro of s use sev eral results from the recen tly dev elop ed F ourier analytic approach to sections of conv ex b o dies; see [K10]. In Section 2 , w e collect necessary definitions and results related to this approac h. F or other results related to the Busemann-P ett y pro blem see [BZ ], [BFM], [K5], [K9 ], [K YY ], [Mi], [Ru], [RZ], [Y1], [Y2], [Zv1], [Zv2]. 2. Elements of the F ourier ap pro ach to sections Our main to ol is t he F ourier transform of distributions. As usual, w e denote by S ( R n ) the Sc hw artz space of rapidly decreasing infin- itely differen tiable functions (test functions) in R n , and S ′ ( R n ) is the space of distributions o v er S ( R n ) . The F ourier transform ˆ f o f a dis- tribution f ∈ S ′ ( R n ) is defined b y h ˆ f , φ i = h f , ˆ φ i f or ev ery test func- tion φ. A distribution is called ev en homogeneous o f degree p ∈ R if h f ( x ) , φ ( x/α ) i = | α | n + p h f , φ i f or ev ery test function φ and eve ry α ∈ R , α 6 = 0 . The F ourier tr a nsform of an ev en homogeneous dis- tribution of degree p is an eve n homogeneous distribution of degree − n − p. A distribution f is called p ositive definite if, f o r ev ery test function φ, h f , φ ∗ φ ( − x ) i ≥ 0 . This is equiv alen t to ˆ f b eing a p ositiv e distribution in the sense that h ˆ f , φ i ≥ 0 for eve ry non-negative test function φ. A compact set K in R n is called a star b o dy if ev ery straig h t line through the origin crosses the b oundary at exactly tw o points different from the orig in, and the b oundary of K is con tinuous in the sense that the Minkowski functional of K defined b y k x k K = min { a ≥ 0 : x ∈ aK } 4 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU is a con tin uous function o n R n . If in addition K is origin symmetric a nd con ve x, then t he Mink owsk i f unctional is a norm on R n . If ξ ∈ S n − 1 , then ρ K ( ξ ) = k ξ k − 1 K is the ra dius of K in the direction ξ . A simple calculation in p olar co ordinat es giv es the following p o lar formula for the volume : n V ol n ( K ) = n Z R n χ ( k x k K ) dx = Z S n − 1 k ξ k − n K dξ , where χ is the indicator function o f t he in terv a l [0 , 1] . W e say that a star b o dy K in R n is k -smo oth ( infinitely smo oth) if the restriction of k x k K to the sphere S n − 1 b elongs to t he class C k ( S n − 1 ) ( C ∞ ( S n − 1 )) of k times con tin uously differen tiable (infinitely differen- tiable) functions on the sphere. It is w ell-kno wn that one can approxi- mate a n y conv ex b o dy in R n in the r a dial metric d ( K , L ) = sup ξ ∈ S n − 1 | ρ K ( ξ ) − ρ L ( ξ ) | b y a sequence of infinitely smo oth conv ex b o dies. This can b e prov ed b y a simple con volution argument (see for example [Sc h , Th. 3.3.1 ]). It is also easy to se e that an y conv ex b o dy in R 2 n in v ariant with res p ect to all R θ can b e appro ximated in the radial metric b y a sequence o f infinitely smo oth conv ex b o dies inv ariant with resp ect to all R θ . This follo ws from the same con volution argumen t, b ecause inv ariance with resp ect to R θ is preserv ed under con v olutions. As pro ve d in [K10, Lemm a 3.16], if K is an infinitely smo oth origin symmetric star b o dy in R n and 0 < p < n then the F ourier transform of the distribution k x k − p K is a homogeneous function of degree − n + p on R n , whose restriction to the sphere is infinitely smo oth. W e use a v ersion o f Parse v al’s formu la on t he sphere established in [K5] (see also [K10, Lemma 3 .2 2]): Prop osition 1. L et K an d L b e infinitely smo oth origin s ymm etric star b o dies in R n and 0 < p < n. Then Z S n − 1  k x k − p K  ∧ ( ξ )  k x k − n + p L  ∧ ( ξ ) dξ = (2 π ) n Z S n − 1 k x k − p K k x k − n + p L dx. The class es of k - intersec tion b o dies were introduced in [K5], [K8] as follo ws. Let 1 ≤ k < n, and let D and L b e o rigin symmetric star b o dies in R n . W e sa y that D is a k -interse ction b o dy of L if for eve ry ( n − k )-dimensional subspace H of R n V ol k ( D ∩ H ⊥ ) = V ol n − k ( L ∩ H ) . COMPLEX BUSEMANN-PETTY PROBLEM 5 More generally , w e say that an origin symmetric star b o dy D in R n is a k -interse ction b o dy if there ex ists a finite Borel measure µ on S n − 1 so tha t for eve ry ev en test function φ ∈ S ( R n ) , Z R n k x k − k D φ ( x ) dx = Z S n − 1  Z ∞ 0 t k − 1 ˆ φ ( tξ ) dt  dµ ( ξ ) . Note that k -in tersection b o dies of star b o dies ar e those k -in tersection b o dies for whic h the measure µ has a contin uous strictly p ositiv e den- sit y; see [K8 ] o r [K10, p. 77]. When k = 1 w e get the class of in tersec- tion bo dies introduced b y Lut w ak in [Lu]. A more general concept of embedding in L − p w as introduced in [K7]. Let D b e an origin symmetric star b ody in R n , and X = ( R n , k · k D ) . F or 0 < p < n, w e sa y that X em b eds in L − p if there ex ists a finite Borel me asure µ on S n − 1 so that, for eve ry ev en test function φ Z R n k x k − p D φ ( x ) dx = Z S n − 1  Z R | z | p − 1 ˆ φ ( z θ ) dz  dµ ( θ ) . Ob viously , an origin sy mmetric star b o dy D in R n is a k -inters ection b o dy if and only if the space ( R n , k · k D ) em b eds in L − k . In this art icle w e use em b eddings in L − p only to state some results in contin uous form; for more applications of this conce pt, see [K10, Ch. 6]. Em b eddings in L − p and k -intersec tion b o dies admit a F o urier ana- lytic c haracterization that w e are going to use throughout this a rticle: Prop osition 2. ( [K8] , [K10, Th. 6.16] ) L et D b e an origin symmetric star b o dy in R n , 0 < p < n. The sp ac e ( R n , k · k D ) emb e ds in L − p if and only if the function k x k − p D r epr e sents a p ositive definite distribution on R n . In p articular, D is a k -interse c tion b o dy if and only if k x k − k D is a p ositive definite distribution on R n . It w as prov ed in [K6] (see also [K10, Corolla ry 4.9]) that ev ery n - dimensional normed space em b eds in L − p for eac h p ∈ [ n − 3 , n ) . In particular, ev ery origin symmetric con vex b o dy in R n is a k - in tersection b o dy f or k = n − 3 , n − 2 , n − 1 . On the other hand, the spaces ℓ n q , q > 2 do no t em b ed in L − p if 0 < p < n − 3 , hence, the unit balls of these spaces a re not k -in tersection b o dies if k < n − 3; see [K3], [K10, Theorem 4.1 3 ]. W e are g oing to use a generalization of the latter result, the so-called second deriv ative test for k -in tersection b o dies and em b eddings in L − p , which w as first prov ed fo r in tersection bo dies in [K4] and then generalized in [K10, Theorems 4.19, 4.21]. Recall that for normed spaces X and Y and q ∈ R , q ≥ 1 , the q - sum ( X ⊕ Y ) q of X and Y is defined as the space of pairs { ( x, y ) : x ∈ X , y ∈ Y } with 6 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU the norm k ( x, y ) k = ( k x k q X + k y k q Y ) 1 /q . Prop osition 3. L et n ≥ 3 , k ∈ N ∪ { 0 } , q > 2 and let Y b e a finite dimensional no rm e d sp ac e of dimens ion gr e ater or e qual to n. Then the q -sum of R and Y do es not emb e d in L − p with 0 < p < n − 2 . In p articular, this dir e ct sum is not a k -interse ction b o dy for any 1 ≤ k < n − 2 . Let 1 ≤ k < n and let H b e an ( n − k )-dimensional subspace of R n . Fix an y o rthonormal basis e 1 , ..., e k in the orthogonal subspace H ⊥ . F or a conv ex b o dy D in R n , define the ( n − k )- dimensional parallel section function A D ,H as a function on R k suc h that A D ,H ( u ) = V ol n − k ( D ∩ { H + u 1 e 1 + ... + u k e k } ) = Z { x ∈ R n :( x,e 1 )= u 1 ,..., ( x, e k )= u k } χ ( k x k D ) dx, u ∈ R k . (2) Let | · | 2 b e the Euclidean norm on R k . F or ev ery q ∈ C , the v a lue of the distribution | u | − q − k 2 / Γ( − q / 2) on a test function φ ∈ S ( R k ) can b e defined in t he usual w a y (see [GS, p.71]) and represen ts an entire function of q ∈ C . If D is infinitely smo oth, the function A D ,H is infinitely differentiable at the origin (see [K10, Lemma 2.4]), and the same regularization pro cedure can b e applied to define the action of these distributions on the function A D ,H . The function q 7→ D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E (3) is an en tire function of q ∈ C . In particular, if q < 0 D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E = 1 Γ( − q / 2) Z R k | u | − q − k 2 A D ,H ( u ) du. If q = 2 m, m ∈ N ∪ { 0 } , then D | u | − q − k 2 Γ( − q / 2)    q =2 m , A D ,H ( u ) E = ( − 1) m | S k − 1 | 2 m +1 k ( k + 2) ... ( k + 2 m − 2) ∆ m A D ,H (0) , (4) where | S k − 1 | = 2 π k / 2 / Γ( k / 2) is the surface area o f the unit sphere S k − 1 in R k , and ∆ = P k i =1 ∂ 2 /∂ u 2 i is the k -dimensional Laplace op erator (for details, see [GS, p.71- 74]). Since the b o dy D is origin- symmetric, the COMPLEX BUSEMANN-PETTY PROBLEM 7 function A D ,H is ev en, and for 0 < q < 2 w e ha ve (see also [K10, p. 49]) D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E = 1 Γ( − q / 2) Z S n − 1  Z ∞ 0 A D ,H ( tθ ) − A D ,H (0) t 1+ q dt  dθ . (5) Note that the function (3) is eq ual ( up to a constan t) to the fractional p o w er of the Lapla cian ∆ q / 2 A D ,H . The follo wing prop osition w as pro ved in [K8 , Th. 2]. W e reproduce the pro of here for the sake of completeness. W e use a we ll-kno wn form ula (see for ex ample [G S, p. 76]): for any v ∈ R k and q < − k + 1 , ( v 2 1 + ... + v 2 k ) ( − q − k ) / 2 = Γ( − q / 2) 2Γ(( − q − k + 1) / 2) π ( k − 1) / 2 Z S k − 1 | ( v , u ) | − q − k du. (6) Prop osition 4. L et D b e a n infinitely s m o oth o ri g i n symmetric c onvex b o dy in R n and 1 ≤ k < n. Then for every ( n − k ) -dim ensional subsp ac e H o f R n and a n y q ∈ R , − k < q < n − k , D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E = 2 − q − k π − k / 2 Γ(( q + k ) / 2)( n − q − k ) Z S n − 1 ∩ H ⊥  k x k − n + q + k D  ∧ ( θ ) d θ . (7) A lso for every m ∈ N ∪ { 0 } , m < ( n − k ) / 2 , ∆ m A D ,H (0) = ( − 1) m 2 k π k ( n − 2 m − k ) Z S n − 1 ∩ H ⊥ ( k x k − n +2 m + k D ) ∧ ( η ) d η , (8) wher e, as b efor e, ∆ is the L aplacian on R k . Pro of : Let first q ∈ ( − k , − k + 1) . Then, D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E = 1 Γ( − q / 2) Z R k | u | − q − k 2 A D ,H ( u ) du. Using the expression (2) for the f unction A D ,H , writing the integral in p olar co ordinates and then using (6 ), w e see that the righ t- hand side of the latt er equation is equal to 1 Γ( − q 2 ) Z R n  ( x, e 1 ) 2 + ... + ( x, e k ) 2 ) ( − q − k ) / 2 χ ( k x k D ) dx = 8 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU 1 Γ( − q 2 )( n − q − k ) Z S n − 1  ( θ , e 1 ) 2 + ... + ( θ , e k ) 2  ( − q − k ) / 2 k θ k − n + q + k D dθ = 1 2Γ( − q − k +1 2 ) π k − 1 2 ( n − q − k ) × Z S n − 1 k θ k − n + q + k D Z S k − 1   ( k X i =1 u i e i , θ )   − q − k du ! dθ = 1 2Γ( − q − k +1 2 ) π k − 1 2 ( n − q − k ) × Z S k − 1 Z S n − 1 k θ k − n + q + k D   ( k X i =1 u i e i , θ )   − q − k dθ ! du. (9) Let us sho w that the function under the in tegral o v er S k − 1 is the F ourier transform of k x k − n + q + k D at t he p oin t P u i e i . F or any ev en test function φ ∈ S ( R n ) , using t he w ell-kno wn connection b et w een the F ourier and Radon transforms (see [K 10, p. 27]) and the expression for the F ourier transform o f the distribution | z | q + k − 1 (see [K10, p. 38]), w e get h ( k x k − n + q + k D ) ∧ , φ i = hk x k − n + q + k D , ˆ φ i = Z R n k x k − n + q + k D ˆ φ ( x ) dx = Z S n − 1 k θ k − n + q + k D  Z ∞ 0 z q + k − 1 ˆ φ ( z θ ) dz  dθ = 1 2 Z S n − 1 k θ k − n + q + k D D | z | q + k − 1 , ˆ φ ( z θ ) E dθ = 2 q + k √ π Γ(( q + k ) / 2) 2Γ(( − q − k + 1) / 2) Z S n − 1 k θ k − n + q + k D D | t | − q − k , Z ( y, θ )= t φ ( y ) dy E dθ = 2 q + k √ π Γ(( q + k ) / 2 ) 2Γ(( − q − k + 1) / 2) Z R n  Z S n − 1 | ( θ , y ) | − q − k k θ k − n + q + k D dθ  φ ( y ) dy . Since φ is an arbitra ry test function, this prov es that, fo r ev ery y ∈ R n \ { 0 } ,  k x k − n + q + k D  ∧ ( y ) = 2 q + k √ π Γ(( q + k ) / 2) 2Γ(( − q − k + 1) / 2) Z S n − 1 | ( θ , y ) | − q − k k θ k − n + q + k D dθ . T ogether with (9), the latter equalit y sho ws that D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E (10) COMPLEX BUSEMANN-PETTY PROBLEM 9 = 2 − q − k π − k / 2 Γ(( q + k ) / 2 )( n − q − k ) Z S n − 1 ∩ H ⊥  k x k − n + q + k D  ∧ ( θ ) dθ , b ecause in our notatio n S k − 1 = S n − 1 ∩ H ⊥ . W e hav e pro v ed (10) under the assumption tha t q ∈ ( − k , − k + 1) . Ho we v er, b oth sides of (10) a r e analytic functions of q ∈ C in the domain where − k < R e ( q ) < n − k. This implies that the equality (10) holds for ev ery q from this domain (see [K10 , p. 61] for the details o f a similar arg umen t). Putting q = 2 m, m ∈ N ∪ { 0 } , m < ( n − k ) / 2 in (10) and applying (4) and the fact that Γ( x + 1) = x Γ( x ), w e get the sec ond form ula. ✷ Brunn’s theorem (see for example [K10, Th. 2.3]) states that the cen tral hyperplane section of an origin symmetric conv ex b o dy has maximal ( n − 1)-dimensional v olume among all h yp erplane sections p erp endicular to a given direc tion. This implies the follo wing Lemma 1. If D is a 2 -sm o oth origin symmetric c o n vex b o dy in R n , then the function A D ,H is twic e differ entiable at the origin and ∆ A D ,H (0) ≤ 0 . Besides that for any q ∈ (0 , 2) , D | u | − q − k 2 Γ( − q / 2) , A D ,H ( u ) E ≥ 0 . Pro of : Differen tiability follo ws from [K10, Lemma 2.4]. Applyin g Brunn’s theorem to the b o dies D ∩ span( H , θ ) , θ ∈ S n − 1 ∩ H ⊥ , we see that the function t 7→ A D ,H ( tθ ) has maxim um at zero. Therefore, the in terior in tegr a l in (5) is negativ e, but Γ( − q / 2) < 0 fo r q ∈ (0 , 2) , whic h implies the second statemen t. T he first inequality also follows from the fact that each of the functions t 7→ A D ,H ( te j ) , j = 1 , ..., k has maxim um at the origin. ✷ W e often use Lemm a 4.10 fro m [K10] for the purp ose of approxima- tion b y infinitely smo oth b o dies. F or con ven ience, let us fo rm ulate this lemma: Lemma 2. ( [K10, Lemma 4.10 ] ) L et 1 ≤ k < n. Supp ose that D is an origin-symmetric c on v e x b o dy in R n that is not a k -interse ction b o dy. Then ther e exis ts a se quenc e D m of ori g i n -symmetric c on v ex b o dies so that D m c onver ges to D in the r adial me tric, e ach D m is infinitely sm o oth, has strictly p ositive curvatur e and e ach D m is not a k -interse ction b o dy. 10 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU If in addition D is in v ariant with respect to R θ , one can c ho ose D m with the same prop erty . 3. Connection with interse ction bodies W e now return to the complex case. The fo llowing simple observ a tion is crucial for applications of the F ourier me tho ds to con v ex b o dies in the complex case: Lemma 3. Supp ose that K is a n origin-symme tric infinitely smo oth invariant with r esp e ct to al l R θ star b o dy in R 2 n . Then for every 0 < p < 2 n and ξ ∈ S 2 n − 1 the F ourier tr ansform of the distribution k x k − p K is a c on stant function on S 2 n − 1 ∩ H ⊥ ξ . Pro of : By [K10, Lemma 3.1 6], the F ourier tra nsform of k x k − p K is a con tin uous function outside of the origin in R 2 n . The function k x k K is inv ariant with respect to all R θ , s o by the connection b et w een the F ourier transform of distributions and linear tr a nsformations, the F ourier t r a nsform of k x k − p K is also inv ariant with respect to all R θ . Re- call that the tw o-dimensional space H ⊥ ξ is spanned by v ectors ξ and ξ ⊥ (see the In tro duction). Ev ery v ector in S 2 n − 1 ∩ H ⊥ ξ is the image of ξ under one of t he co ordinate-wise rotat io ns R θ , so the F ourier transform of k x k − p K is a constant function on S 2 n − 1 ∩ H ⊥ ξ . ✷ Of course, this argument also applies t o the F ourier transform o f an y distribution of the form h ( k x k K ) . Similarly to the real case (see [K1], [K10, Theorem 3.8]), one can express the v o lume of hyperplane sections in terms of the F ourier tr a ns- form. Theorem 1. L et K b e an in fi nitely smo oth origin symmetric inv a riant with r e s p e ct to R θ c onvex b o dy in R 2 n , n ≥ 2 . F or every ξ ∈ S 2 n − 1 , we have V ol 2 n − 2 ( K ∩ H ξ ) = 1 4 π ( n − 1)  k x k − 2 n +2 K  ∧ ( ξ ) . Pro of : Let us fix ξ ∈ S 2 n − 1 . W e apply formula (8 ) with 2 n in place of n , H = H ξ , k = 2 , m = 0 . W e get V ol 2 n − 2 ( K ∩ H ξ ) = A K,H ξ (0) = 1 8 π 2 ( n − 1) Z S 2 n − 1 ∩ H ⊥ ξ  k x k − 2 n +2 K  ∧ ( η ) dη . By Lemma 3, the function under the integral in the right hand side is constan t on the circle S 2 n − 1 ∩ H ⊥ ξ . Since ξ ∈ H ⊥ ξ , the in tegral is eq ual to 2 π  k x k − 2 n +2 K  ∧ ( ξ ) . COMPLEX BUSEMANN-PETTY PROBLEM 11 ✷ The conne ction b et wee n the complex Busemann-P etty problem and in tersection b o dies is as follo ws: Theorem 2. The answer to the c omplex Busemann-Pe tty pr oblem in C n is affi rm ative if and only if every origin symmetric in variant with r esp e c t to al l R θ c onvex b o dy in R 2 n is a 2 -interse ction b o dy. This theorem will follo w fr o m the next t w o lemmas. Note that, since w e can appro ximate the b o dy K in the radia l metric from inside b y infinitely smooth con vex bo dies in v aria n t with respect to all R θ , and also approximate L from o utside in the same w a y , we can argue that if the answ er to the complex Busemann-P etty problem is affirmativ e f o r infinitely smoo th b o dies K a nd L then it is affirmativ e in general. Lemma 4. L et K and L b e infi n itely smo oth invari a n t with r esp e ct to R θ c onvex b o dies in R 2 n so that K is a 2 -interse ction b o dy a n d, for every ξ ∈ S 2 n − 1 , V ol 2 n − 2 ( K ∩ H ξ ) ≤ V o l 2 n − 2 ( L ∩ H ξ ) . Then V ol 2 n ( K ) ≤ V ol 2 n ( L ) . Pro of : By [K10, Lemma 3.16], the F o ur ier transforms of the distri- butions k x k − 2 n +2 K , k x k − 2 n +2 L and k x k − 2 K are contin uous functions outside of the origin in R 2 n . By Theorem 1 and Prop o sition 2 , the conditions of the lemma imply that for eve ry ξ ∈ S 2 n − 1 ,  k x k − 2 n +2 K  ∧ ( ξ ) ≤  k x k − 2 n +2 L  ∧ ( ξ ) and  k x k − 2 K  ∧ ( ξ ) ≥ 0 . Therefore, Z S 2 n − 1  k x k − 2 n +2 K  ∧ ( ξ )  k x k − 2 K  ∧ ( ξ ) d ξ ≤ Z S 2 n − 1  k x k − 2 n +2 L  ∧ ( ξ )  k x k − 2 K  ∧ ( ξ ) d ξ . No w we apply Parsev al’s formula on t he sp here, Prop osition 1, to re- mo ve the F ourier transforms in the latter inequalit y and then use the p olar form ula for the volume a nd H¨ older’s inequalit y: 2 n V ol 2 n ( K ) = Z S 2 n − 1 k x k − 2 n K dx ≤ Z S 2 n − 1 k x k − 2 n +2 L k x k − 2 K dx ≤  Z S 2 n − 1 k x k − 2 n L dx  n − 1 n  Z S 2 n − 1 k x k − 2 n K dx  1 n 12 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU = (2 n V ol 2 n ( L )) n − 1 n (2 n V ol 2 n ( K )) 1 n , whic h giv es the res ult. ✷ Lemma 5. Supp ose that ther e exists an origin symmetric invariant with r esp e ct to al l R θ c onvex b o dy L in R 2 n which is not a 2 -interse ction b o dy. Then one c an p erturb L twic e to c onstruct other origin symm etric invariant with r esp e ct to R θ c onvex b o dies L ′ and K in R 2 n such that for every ξ ∈ S 2 n − 1 , V ol 2 n − 2 ( K ∩ H ξ ) ≤ V ol 2 n − 2 ( L ′ ∩ H ξ ) , but V ol 2 n ( K ) > V ol 2 n ( L ′ ) . Pro of : W e can assume that the b o dy L is infinitely smo oth and has strictly p ositiv e curv ature. In fact, approximating L in the radial met- ric b y infinitely smo oth in v a rian t with res p ect to all R θ con ve x b o dies with strictly p ositiv e curv ature, w e get by Lemma 2 that approximating b o dies can not all b e 2-in tersection b o dies. So t here exists an infinitely smo oth in v aria n t with respect to all R θ con ve x b o dy L ′ with stric tly p ositiv e curv ature that is not a 2-in tersection b o dy . No w a s L is infinitely smo oth, b y [K10, Lemma 3.16 ], the F ourier transform of k x k − 2 L is a contin uous function outside of the origin in R 2 n . The bo dy L is not a 2-in tersection bo dy , so b y Prop osition 2 , the F ourier transform  k x k − 2 L  ∧ is negat ive on some o p en subset Ω of the sphere S 2 n − 1 . Since L is inv arian t with resp ect to rot a tions R θ , we can assume that the set Ω is also in v ariant with resp ect to rotations R θ . This allows us to c ho o se an ev en non-nega t ive in v ariant with resp ect to rot a tions R θ function f ∈ C ∞ ( S 2 n − 1 ) whic h is supp orted in Ω . Extend f to an ev en homogeneous f unction f ( x/ | x | 2 ) | x | − 2 2 of degree -2 on R 2 n . By [K10, Lemma 3.16], the F ourier transform of this extension is an ev en homogeneous f unction of degree -2n+2 o n R 2 n , whose restriction to the sphere is infinitely smo oth:  f ( x/ | x | 2 ) | x | − 2 2  ∧ ( y ) = g ( y / | y | 2 ) | y | − 2 n +2 2 , where g ∈ C ∞ ( S 2 n − 1 ) . By the connection b etw een the F ourier trans- form and linear transforma t io ns, the function g is also inv ariant with resp ect to rotations R θ . Define a bo dy K in R 2 n b y k x k − 2 n +2 K = k x k − 2 n +2 L − ǫg ( x/ | x | 2 ) | x | − 2 n +2 2 . (11) COMPLEX BUSEMANN-PETTY PROBLEM 13 F or small enough ǫ the b o dy K is con vex . This essen tia lly follo ws fro m a simple t w o- dimensional argumen t : if h is a strictly concav e function on an in terv al [ a, b ] and u is a t wice differen tiable function on [ a, b ], then for small ǫ the f unction h + ǫu is a lso conca ve . Note that here w e use the condition tha t L has strictly p ositiv e curv ature. Besides tha t, the b o dy K is in v ariant with resp ect t o rotations R θ b ecause so are the b o dy L and the function g . W e can now c ho o se ǫ so that K is an origin sym metric in v aria n t with respect to all R θ con ve x b o dy in R 2 n . Let us prov e that the b o dies K and L pro vide the necessary coun- terexample. W e apply the F ourier transform to b oth sides of (11). By definition o f the function g and since f is non- negativ e, we get that for ev ery ξ ∈ S 2 n − 1  k x k − 2 n +2 K  ∧ ( ξ ) =  k x k − 2 n +2 L  ∧ ( ξ ) − (2 π ) 2 n ǫf ( ξ ) ≤  k x k − 2 n +2 L  ∧ ( ξ ) . By Theorem 1 , this means that for eve ry ξ V ol 2 n − 2 ( K ∩ H ξ ) ≤ V o l 2 n − 2 ( L ∩ H ξ ) . On the other hand, the function f is p ositiv e only where  k x k − 2 L  ∧ is negativ e, so Z S 2 n − 1  k x k − 2 n +2 K  ∧ ( ξ )  k x k − 2 L  ∧ ( ξ ) d ξ = Z S 2 n − 1  k x k − 2 n +2 L  ∧ ( ξ )  k x k − 2 L  ∧ ( ξ ) d ξ − (2 π ) 2 n ǫ Z S 2 n − 1  k x k − 2 L  ∧ ( ξ ) f ( ξ ) dξ > Z S 2 n − 1  k x k − 2 n +2 L  ∧ ( ξ )  k x k − 2 L  ∧ ( ξ ) d ξ . The end of the pro of is similar to that of t he previous lemma - w e apply P arsev al’s formula to remo v e F ourier transforms and then use H¨ older’s inequalit y and the p olar formula for the volume to get V ol n ( K ) > V ol n ( L ) . ✷ 4. The solution of the pro blem It is kno wn (see [K6] or [K10 , Corollary 4.9 ] plus Prop osition 2) that for ev ery origin symmetric con ve x b o dy K in R 2 n , n ≥ 2 the space ( R 2 n , k · k K ) embeds in L − p for eac h p ∈ [2 n − 3 , 2 n ) , or, in other w o r ds, ev ery orig in- symmetric conv ex b o dy in R 2 n is a (2 n − 3)-, (2 n − 2)- a nd (2 n − 1)-inters ection b o dy . On the other hand, for q > 2 the unit ball of the real space ℓ 2 n q is not a (2 n − 4)-in tersection b o dy , and, mor eov er, 14 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU R 2 n pro vided with the no r m of this space do es not em b ed in L − p with p < 2 n − 3 (see [K3] or [K10, Th. 4.13]). No w w e ha v e to find out what happ ens if we consider conv ex b odies in v ariant with resp ect to all R θ . It immediately follows from the second deriv ative tes t ([K10, Th. 4 .19 and 4.21] ; see Corolla r y 4 b elo w) that for q > 2 the complex space ℓ n q do es not embed in L − p with p < 2 n − 4 , whic h means that the unit ball B n q of this space (whic h is inv ariant with resp ect t o all R θ ) is no t a k -inte rsection b o dy with k < 2 n − 4 . The only question that remains op en is what happ ens in the in terv al p ∈ [2 n − 4 , 2 n − 3) . The follo wing result answ ers this question. Theorem 3. L et n ≥ 3 . Every origin symmetric in variant with r esp e ct to R θ c onvex b o dy K in R 2 n is a (2 n − 4) -interse ction b o d y. Mor e over, the sp ac e ( R 2 n , k · k K ) emb e ds in L − p for every p ∈ [2 n − 4 , 2 n ) . If n = 2 the sp ac e ( R 2 n , k · k K ) emb e ds in L − p for every p ∈ (0 , 4 ) . Pro of : By Lemma 2 , it is enough t o pro ve the result in the case where K is infinitely smo oth. Fix ξ ∈ S 2 n − 1 . Let n ≥ 3 . Applying formula (8) and then Lemma 3 with H = H ξ , m = 1 , k = 2 and dime nsion 2 n instead o f n, w e get ∆ A K,H ξ (0) = − 1 8 π 2 ( n − 2) Z S n − 1 ∩ H ⊥ ξ ( k x k − 2 n +4 K ) ∧ ( η ) dη = − 2 π 8 π 2 ( n − 2)  k x k − 2 n +4 K  ∧ ( ξ ) . By Brunn’s theorem (see Lemma 1) ,  k x k − 2 n +4 K  ∧ ( ξ ) ≥ 0 for ev ery ξ ∈ S 2 n − 1 , so k x k − 2 n +4 K is a p ositiv e definite distribution on R 2 n . By Prop osition 2, K is a (2 n − 4)-in tersection bo dy . No w let n ≥ 2 . F or 0 < q < 2, formula (7 ) and Lemma 1 imply that  k x k − 2 n + q +2 K  ∧ ( ξ ) ≥ 0 . By Prop o sition 2, the space ( R 2 n , k · k K ) em b eds in L − 2 n + q +2 , and, using the rang e of q , ev ery suc h space em b eds in L − p , p ∈ (2 n − 4 , 2 n − 2) . As men tioned b efore, these spaces also em b ed in L − p , p ∈ [2 n − 3 , 2 n ) , because so does an y 2 n -dimensional normed space. ✷ W e no w giv e a n example of an o r ig in symmetric in v arian t with re- sp ect to all R θ con ve x b o dy in R 2 n whic h is not a k -in t ersection b o dy for an y 1 ≤ k < 2 n − 4 . Denote b y B q n the unit ball of the complex space ℓ n q considered as a subset of R 2 n : B n q = { ξ ∈ R 2 n : k ξ k q =   ξ 2 11 + ξ 2 12  q / 2 + ... +  ξ 2 n 1 + ξ 2 n 2  q / 2  1 /q ≤ 1 } . COMPLEX BUSEMANN-PETTY PROBLEM 15 If q ≥ 1 then B n q is an origin sy mmetric in v ariant with resp ect t o R θ con ve x b o dy in R 2 n . The next theorem immediately follows from Prop osition 3. Theorem 4. If q > 2 then the sp ac e ( R 2 n , k · k q ) do es no t emb e d i n L − p with 0 < p < 2 n − 4 . In p articular, the b o dy B n q is not a k -interse ction b o dy for any 1 ≤ k < 2 n − 4 . Pro of : The space ( R 2 n , k · k q ) con tains as a subspace the q -sum of R and a (2 n − 2)-dimensional subspace ( R 2 n − 2 , k · k q ) . This q - sum do es not em b ed in L − p , 0 < p < 2 n − 4 b y Prop osition 3. By a result of E.Milman [Mi], the larger space cannot embed in L − p , 0 < p < 2 n − 4 either (the pro of in [Mi ] is only for in tegers p , but it is exactly the same for non- in tegers; note that for the complex Busemann-P etty problem w e need only the second statemen t of the corolla ry , where p is an inte ger). ✷ W e are no w ready to pro v e the main result of this article: Theorem 5. The solution to the c omplex Busemann-Petty pr obl e m in C n is affirm a tive if n ≤ 3 and it is ne gative i f n ≥ 4 . Pro of : B y Theorem 3, ev ery origin symmetric inv arian t with resp ect to R θ con ve x b o dy in R 6 (where n = 3) is a 2 n − 4 = 2-interse ction b o dy , and in R 4 (where n = 2) it is a 2 n − 2 = 2 - in tersection b o dy . The affirmative answ ers fo r n = 3 and n = 2 fo llow now from Theorem 2. If n ≥ 4 then 2 n − 4 > 2 , so b y Theorem 4 the bo dy B n q is not a 2-in tersection b o dy . The negativ e answ er follows from Theorem 2. ✷ Remark 1. The transition b et w een the dimensions n = 3 and n = 4 is due to the f act that conv exit y con trols only deriv ativ es o f the second order. T o see this let us lo ok again at formula (8), whic h w e apply with k = 2 . W e w ant to get informa t ion ab out the F ourier tra nsfor m of k x k − 2 D , so w e need to c ho ose m so that − 2 n + 2 m + 2 = − 2 . If n = 3 then m = 1 , but when n = 4 w e need m = 2 . This means that for n = 3 w e consider ∆ A K,H (0) , wh ic h is alw ays negativ e b y con vex it y , but when n = 4 w e lo o k at ∆ 2 A K,H (0) , whic h is not controlled b y conv exit y and can b e sign-c hanging . One can construct a coun terexample in dimension n = 4 using this argumen t, similarly to ho w it w a s done for the “real” Busemann-Pe tt y problem; see [K1 0 , Coro llary 4.4]. 16 A. KOLDOBSKY, H. K ¨ ONIG, AND M. ZYMONOPOU LOU Remark 2. Applying Theorem 3 to n = 2 we get that eve ry t wo-dimens ional complex normed space (whic h is a 4- dimensional real normed space) em b eds in L − p for ev ery p ∈ [ − 1 , 0) . By [KKYY, Th. 6.4], this implies that ev ery suc h space em b eds isometrically in L 0 . The concept o f em b edding in L 0 w as introduced in [KK YY]: a normed space ( R n , k · k ) em b eds in L 0 if there exist a probabilit y measure µ on S n − 1 and a constant C so that for ev ery x ∈ R n , x 6 = 0 log k x k = Z S n − 1 log | ( x, ξ ) | dµ ( ξ ) + C . W e ha v e Theorem 6. Every two-dimensional c omplex norme d sp ac e emb e ds in L 0 . 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Ann. 331 (2 005), 867- 8 87. Alexander Koldobsky, Dep ar tment of Ma thema tics, University of Missouri, Columbia, MO 65211 , USA E-mail addr ess : koldobsk@ math.m issouri.edu Hermann K ¨ onig, Ma thema tisches Seminar, Christian-Albrechts-Universitt Kiel, Ludewig-Meyn Str. 4, D-24098 Kiel E-mail addr ess : hkoenig@m ath.un i-kiel.de Marisa Zymonopoulou, Dep ar tment of Ma thema tics, University of Missouri, Columbia, MO 65211 , USA E-mail addr ess : marisa@ma th.mis souri.edu