BRUNN-MINKOWSKI INEQUALITY FOR L p -MIXED INTERSECTION BODIES

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1 Miskolc Mathematical Notes HU e-issn Vol. 18 (2017), No. 1, pp DOI: /MMN BRUNN-MINKOWSKI INEQUALITY FOR L p -MIXED INTERSECTION BODIES CHANG-JIAN HAO AND MIHÁLY BENCE Received 02 October, 2013 Abstract. In this paper, we establish L p -Brunn-Minkowski inequality for dual Quermassintegral of L p -mixed intersection bodies. As application, we give the well-known Brunn-Minkowski inequality for mixed intersection bodies Mathematics Subject Classification: 52A40 Keywords: the Brunn-Minkowski inequality, L p -dual mixed volumes, L p -mixed intersection bodies 1. INTRODUCTION The intersection operator and the class of intersection bodies were defined by Lutwak [9]. The closure of the class of intersection bodies was studied by Goody, Lutwak, and Weil [5]. The intersection operator and the class of intersection bodies played a critical role in hang [12] and Gardner [2] on the solution of the famous Busemann-Petty problem (See also Gardner, Koldobsky, Schlumprecht [4]). As Lutwak [9] shows (and as is further elaborated in Gardner s book [3]), there is a kind of duality between projection and intersection bodies. Consider the following illustrative example: It is well known that the projections (onto lower dimensional subspaces) of projection bodies are themselves projection bodies. Lutwak conjectured the dualiy : When intersection bodies are intersected with lower dimensional subspaces, the results are intersection bodies (within the lower dimensional subspaces). This was proven by Fallert, Goodey and Weil [1]. In [7] (see also [10] and [8]), Lutwak introduced mixed projection bodies and proved the following Brunn-Minkowski inequality for mixed projection bodies: Theorem 1. If K;L 2 K n and 0 i < n, then W i.p.k C L// 1=.n i/.n 1/ W i.pk/ 1=.n i/.n 1/ C W i.pl/ 1=.n i/.n 1/ ; (1.1) with equality if and only if K and L are homothetic. The first author was supported in part by the National Natural Sciences Foundation of China, Grant No c 2017 Miskolc University Press

2 508 CHANG-JIAN HAO AND MIHÁLY BENCE Where, K n denotes the set of convex bodies in R n. W i.k/ D V.K;:::;K;B;:::;B/ ƒ ƒ n i i denotes the classical Quermassintegral of convex body K. PK denotes the projection body of convex body K. In 2008, the Brunn-Minkowski inequality for mixed intersection bodies was established as follows [13]. Theorem 2. If K;L 2 ' n, 0 i < n, then QW i.i.k QCL// 1=.n i/.n 1/ QW i.ik/ 1=.n i/.n 1/ C QW i.il/ 1=.n i/.n 1/ ; (1.2) Where, ' n denotes the set of star bodies in R n. Associated with a compact subset K of R n, which is star-shaped with respect to the origin, is its radial function.k;/ W S n 1! R; defined for u 2 S n 1, by.k;u/ D Maxf 0 W u 2 Kg: If.K;/ is positive and continuous, K will be called a star body. Moreover, IK denotes the intersection body of star body K and the sum QC denotes the radial Minkowski sum and QW i.k/ D QV.K;:::;K;B;:::;B/ denotes the classical dual ƒ ƒ n i i Quermassintegral of star body K. In 2006, Haberl and Ludwig [6] introduced L p -intersection bodies(p < 1). For K 2 P0 n, where P 0 n denotes the set of convex polytopes in Rn that contain the origin in their interiors. The star body Ip C K is defined for u 2 S n 1 by.ip C K;u/p D ju xj p dx; (1.3) K\u C where u C D fx 2 R n W ux 0g; and define I p K D Ip C. K/: For p < 1, the centrally symmetric star body I p K D I C p K C I p K is called as the L p intersection body of K. So for u 2 S n 1, p.i p K;u/ D ju xj p dx: (1.4) K The purpose of this paper is to establish Brunn-Minkowski inequality for L p - mixed intersection bodies as follows Theorem 3. If K;L 2 ' n, and 0 i < n; then for p < 1 QW i.i p.k QCL// 1=.n i/.n 1/ QW i.i p K/ 1=.n i/.n 1/ C QW i.i p L/ 1=.n i/.n 1/ ; (1.5) Where, I p K denotes the above L p -intersection body of star body K which was defined by Haberl and Ludwig [6].

3 BRUNN-MINKOWSKI INEQUALITY 509 Remark 1. Let p! 1 in (1.5), (1.5) changes to (1.2). To prove Theorem 3, the paper first introduce a new notion L p -dual mixed volumes, then generalize Haberl and Ludwig s L p -intersection bodies to L p -mixed intersection bodies (p < 1). Moreover, we use a new way which is different from the way of [13]. 2. PRELIMINARIES The setting for this paper is n-dimensional Euclidean space R n.n > 2/. Let C n denote the set of non-empty convex figures(compact, convex subsets) and K n denote the subset of C n consisting of all convex bodies (compact, convex subsets with nonempty interiors) in R n. We reserve the letter u for unit vectors, and the letter B is reserved for the unit ball centered at the origin. The surface of B is S n 1. For u 2 S n 1, let E u denote the hyperplane, through the origin, that is orthogonal to u. We will use K u to denote the image of K under an orthogonal projection onto the hyperplane E u. We use V.K/ for the n-dimensional volume of convex body K. The support function of K 2 K n, h.k;/, defined on R n by h.k;/ D Maxfx y W y 2 Kg: Let ı denote the Hausdorff metric on K n ; i.e., for K;L 2 K n ; ı.k;l/ D jh K h L j 1 ; where jj 1 denotes the sup-norm on the space of continuous functions, C.S n 1 /: Let ı Q denote the radial Hausdorff metric, as follows, if K;L 2 ' n, then Qı.K;L/ D j K L j 1 : 2.1. L p -dual mixed volumes We define vector addition QC on R n, which we shall call the radial addition, as follows. For any x 1 ;:::;x r 2 R n, x 1 QC QCx r is defined to be the usual vector sum of x 1 ;:::;x r if they all lie in a 1-dimensional subspace of R n, and as the zero vector otherwise. If K 1 ;:::;K r 2 ' n and 1 ;:::; r 2 R, then the radial Minkowski linear combination, 1 K 1 QC QC r K r ; is defined by 1 K 1 QC QC r K r D f 1 x 1 QC QC r x r W x i 2 K i g: The following property will be used later. If K;L 2 ' n and ; 0.K QCL;/ D.K;/ C.L;/: (2.1) For K 1 ;:::;K r 2 ' n and 1 ;:::; r 0, the volume of the radial Minkowski liner combination 1 K 1 QC QC r K r is a homogeneous nth-degree polynomial in the i [11], V. 1 K 1 QC QC r K r / D X QV i1 ;:::;i n i1 in (2.2) where the sum is taken over all n-tuples.i 1 ;:::;i n / whose entries are positive integers not exceeding r. If we require the coefficients of the polynomial in (2.1.2) to be symmetric in their arguments, then they are uniquely determined. The coefficient QV i1 ;:::;i n is nonnegative and depends only on the bodies K i1 ;:::;K in. It is written as

4 510 CHANG-JIAN HAO AND MIHÁLY BENCE QV.K i1 ;:::;K in / and is called the dual mixed volume of K i1 ;:::;K in : If K 1 D D K n i D K; K n ic1 D D K n D L, the dual mixed volumes is written as QV i.k;l/. The dual mixed volumes QV i.k;b/ is written as QW i.k/. If K i 2 ' n.i D 1;2;:::;n 1/, then the dual mixed volume of K i \ E u.i D 1;2;:::;n 1/ will be denoted by Qv.K 1 \E u ;:::;K n 1 \E u /. If K 1 D ::: D K n 1 i D K and K n i D ::: D K n 1 D L; then Qv.K 1 \ E u ;:::;K n 1 \ E u / is written Qv i.k \ E u ;L \ E u /: If L D B, then Qv i.k \ E u ;B \ E u / is written Qw i.k \ E u /: L p -dual mixed volumes was defined as follows [14]. 1 1=p QV p.k 1 ;:::;K n / D! n p.k 1 ;u/ p.k n ;u/ds.u/ ; p 0; n! n S n 1 (2.3) where K 1 ;:::;K n 2 ' n : If K 1 D ::: D K n 1 i D K and K n i D ::: D K n 1 D L; will write QV p.k;:::;k;l;:::;l/ as QV ƒ ƒ p;i.k;l). If K 1 D ::: D K n D K, will write QV p.k;:::;k/ ƒ n 1 i i n as QV p.k/. If L D B, then write QV p.k;:::;k;b;:::;b/ as QV p;i.k/ and is called L p - ƒ n i ƒ i dual Quermassintegral as follows. 1 QV p;i.k/ D! n n! n S n p.n 1 1=p i/.k;u/ds.u/ ; p 0: (2.4) Remark 2. Apparently, let p D 1, then L p -dual mixed volumes QV p and L p -dual Quermassintegral QV p;i change to the classical dual mixed volumes QV and dual Quermassintegral QW i, respectively. and 2.2. L p -mixed intersection bodies Since [6] v.k \ u C " / D lim ju xj "!0 2 K 1C" dx: (2.5) 1 p.ik; u/ D lim p!1 2 p.i p K;u/; (2.6) that is, the intersection body of K is obtained as a limit of L p intersection bodies of K. Also note that a change to polar coordinates in (2.6) shows that up to a normalization factor p.i p K;u/ equals the Cosine transform of.k;u/ n p. Here, we introduce the L p -mixed intersection bodies of K 1 ;:::;K n 1. It is written as I p.k 1 ;:::;K n 1 /.p < 1/, whose radial function is defined by p.i p.k 1 ;:::;K n 1 /;u/ D 2 1 p Qv p.k 1 \ E u ;:::;K n 1 \ E u /; (2.7)

5 BRUNN-MINKOWSKI INEQUALITY 511 where, Qv p.k 1 \ E u ;:::;K n 1 \ E u / denotes the p-dual mixed volumes of K 1 \ E u ;:::;K n 1 \E u in.n 1/-dimensional space. If K 1 D D K n i 1 D K;K n i D D K n 1 D L, then Qv p.k 1 \ E u ;:::;K n 1 \ E u / is written as Qv p;i.k \ E u;l \ E u /. If L D B, then Qv p;i.k \ E u;l \ E u / is written as Qv p;i.k \ E u/. Remark 3. From the definition, which introduces a new star body, namely the L p -mixed intersection body of n 1 given bodies. From the definition, V p.k 1 ;:::;K n / is continuous function for any K i 2 ' n ; i D 1;2;:::;n; then lim p!1 1 p 2 p.i p.k 1 ;:::;K n 1 /;u/ 1 1=p D lim! n p.k 1 ;u/ p.k n 1 ;u/ds.u/ p!1 n! n S n 1 D 1.K 1 ;u/.k n 1 ;u/ds.u/: n S n 1 On the other hand, by using definition of mixed intersection bodies(see [3] and [14]), we have.i.k 1 ;:::;K n 1 /;u/ D Qv.K 1 \ E u ;:::;K n 1 \ E u / D 1.K 1 ;u/.k n 1 ;u/ds.u/: n S n 1 Hence lim p!1 1 p 2 p.i p.k 1 ;:::;K n 1 /;u/ D.I.K 1 ;:::;K n 1 /;u/: For the L p -mixed intersection bodies, I p.k 1 ;:::;K n 1 /, if K 1 D D K n i 1 D K;K n i D D K n 1 D L, then I p.k 1 ;:::;K n 1 / is written as I p.k;l/ i. If L D B, then I p.k;l/ i is written as I p K i is called the ith L p -intersection body of K. For I p K 0 simply write I p K, this is just the L p -intersection bodies of star body K. The following properties will be used later: If K;L, M;K 1 ;:::;K n 1 2 ' n, and ;; 1 ;:::; n 1 > 0, then where M D.K 1 ;:::;K n 2 /. I p.k QCL;M / D I p.k;m / QCI p.l;m /; (2.8) 3.1. Some Lemmas 3. MAIN RESULTS The following results will be required to prove our main Theorems.

6 512 CHANG-JIAN HAO AND MIHÁLY BENCE Lemma 1. If K;L 2 ' n, 0 i < n;0 j < n 1;i;j 2 N and p < 1, then QW i.i p.k;l/ j / D 1 n i 2 p Qv p;j n 1 p.k \ E.n i/ u;l \ E u / p ds.u/: (3.1) S n 1 From (2.4) and (2.7), identity (3.1) in Lemma 1 easy follows. Lemma 2. If K 1 ;:::;K n 2 ' n, 1 < r n, 0 j < n 1;j 2 N and p 0; then ry QV p.k 1 ;:::;K n / r QV p.k j ;:::;K j ;K rc1 ;:::;K n /; (3.2) ƒ j D1 r with equality if and only if K 1 ;:::;K n are all dilations [14]. From (3.1), (3.2) and in view of Hölder inequality for integral, we obtain Lemma 3. If K;L 2 ' n, 0 i < n, 0 < j < n 1; and p < 1; then QW i.i p.k;l// n 1 QW i.i p K/ n j 1 QW i.i p L/ j ; (3.3) with equality if and only if K and K are dilations Brunn-Minkowski inequality for L p -mixed intersection bodies The Brunn-Minkowski inequality for L p -intersection bodies, which will be established is: If K;L 2 ' n, p < 1 then V.I p.k QCL// 1=n.n 1/ V.I p K/ 1=n.n 1/ C V.I p L/ 1=n.n 1/ ; (3.4) This is just the special case i D 0 of: Theorem 4. If K;L 2 ' n, and 0 i < n; then QW i.i p.k QCL// 1=.n i/.n 1/ QW i.i p K/ 1=.n i/.n 1/ C QW i.i p L/ 1=.n i/.n 1/ ; (3.5) Proof. Let M D.L 1 ;:::;L n 2 /, from (2.1), (2.4), (2.8) and in view of Minkowski inequality for integral, we obtain that QW i.i p.k QCL;M // 1=.n i/ D n 1=.n i/ k.i p.k QCL;M /;u/k n i D n 1=.n i/ k.i p.k;m / QCI p.l;m /;u/k n i D n 1=.n i/ k.i p.k;m /;u/ C.I p.l;m /;u/k n i n 1=.n i/ k.i p.k;m /;u/k n i C k.i p.l;m /;u/k n i D QW i.i p.k;m // 1=.n i/ C QW i.i p.l;m // 1=.n i/ : (3.6) On the other hand, taking L 1 D D L n 2 D K QCL to (3.6) and apply Lemma 2 and Lemma 3, and get

7 BRUNN-MINKOWSKI INEQUALITY 513 QW i.i p.k QCL// 1=.n i/ QW i I p.k;k QCL/ n 2 / 1=.n i/ 1=.n i/ C QW i.i p.l;k QCL/ n 2 / QW i.i p K/ 1=.n 1/.n i/.n 2/=.n 1/.n i/ QW i.i p.k QCL// C QW i.i p L/ 1=.n 1/.n i/ QW i.i p.k QCL//.n 2/=.n 1/.n i/ ; (3.7) with equality if and only if K, L and M D K QCL are dilates, combine this with the equality condition of (3.6), it follows that the condition holds if and only if K and L are dilates. Dividing both sides of (3.7) by QW i.i p.k QCL//.n 2/=.n 1/.n i/, we get the inequality (3.5). The proof is complete. Remark 4. Let i D 0 and p! 1 in (2.6), we get the well-known Brunn-Minkowski inequality for mixed intersection bodies as follows: QV.I.K QCL// 1=n.n 1/ QV.IK/ 1=n.n 1/ 1=n.n 1/ C QV.IL/ REFERENCES [1] H. Fallert, P. Goodey, and W. Weil, Spherical projections and centrally symmetric sets, Advances in Mathematics, vol. 129, no. 2, pp , [2] R. J. Gardner, A positive answer to the busemann-petty problem in three dimensions, Annals of Mathematics, pp , [3] R. J. Gardner, Geometric tomography. Cambridge University Press Cambridge, 1995, vol. 6. [4] R. J. Gardner, A. Koldobsky, and T. Schlumprecht, An analytic solution to the busemann-petty problem on sections of convex bodies, Annals of Mathematics, vol. 149, pp , [5] P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations of classes of convex bodies, Mathematische eitschrift, vol. 222, no. 3, pp , [6] C. Haberl and M. Ludwig, A characterization of lp intersection bodies, International Mathematics Research Notices, vol. 2006, p , [7] E. Lutwak, Mixed projection inequalities, Transactions of the American Mathematical Society, vol. 287, no. 1, pp , [8] E. Lutwak, Volume of mixed bodies, Transactions of the American Mathematical Society, vol. 294, no. 2, pp , [9] E. Lutwak, Intersection bodies and dual mixed volumes, Advances in Mathematics, vol. 71, no. 2, pp , [10] E. Lutwak, Inequalities for mixed projection bodies, Transactions of the American Mathematical Society, vol. 339, no. 2, pp , [11] R. Schneider, Convex bodies: the Brunn Minkowski theory. Cambridge University Press, 2013, no [12] G. hang, A positive solution to the busemann-petty problem in rˆ 4, Annals of Mathematics, vol. 149, pp , [13] C. hao and G. Leng, Brunn minkowski inequality for mixed intersection bodies, Journal of mathematical analysis and applications, vol. 301, no. 1, pp , 2005.

8 514 CHANG-JIAN HAO AND MIHÁLY BENCE [14] C. hao, L p-mixed intersection bodies, Science in China Series A: Mathematics, vol. 51, no. 12, pp , Authors addresses Chang-Jian hao Department of Mathematics, China Jiliang University, Hangzhou , P.R.China address: chjzhao@163.com.com chjzhao@aliyun.com Mihály Bencze Str. H Marmanului 6, S Lacele-NLegyfalu, Jud, Braşov, Romania, Romania address: benczemihaly@yahoo.com benczemihaly@gmail.com