Brunn-Minkowski type inequalities for L p Blaschke- Minkowski homomorphisms

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1 Available online at J. Nonlinear Sci. Appl. 9 (206), Research Article Brunn-Minkowski type inequalities for L p Blaschke- Minkowski homomorphisms Feixiang Chen a,b,, Gangsong Leng a a Department of Mathematics, Shanghai University, Shanghai , China. b School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou , China. Communicated by A. Petrusel Abstract In this paper, the Brunn-Minkowski type inequalities for L p Blaschke-Minkowski homomorphisms and L p radial Minkowski homomorphisms are established. c 206 All rights reserved. Keywords: Brunn-Minkowski inequality, L p Blaschke-Minkowski homomorphisms. 200 MSC: 52A20, 52A40.. Introduction and preliminaries The Brunn-Minkowski inequality is one of the most important geometric inequalities. There is a huge amount of work on its generalizations and on its connections with other areas (see [, 5 7, 6, 8]). The excellent survey article of Gardner [5] gives a comprehensive account of various aspects and consequences of the Brunn-Minkowski inequality. Projection bodies and intersection bodies played a critical role in the solution of the Shephard problem and the Busemann-Petty problem, respectively (see [4]). Through the work of Ludwig [2, 3], projection bodies and intersection bodies were characterized as continuous and GL(n) contravariant valuations. Recently, Schuster [9, 20] introduced the Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski homomorphisms which are more general than the well-known projection body operators and intersection bodies, respectively. In order to state their definition, let K n denote the space of all convex bodies in R n endowed with the Hausdorff topology. A map Φ : K n K n is called a Blaschke-Minkowski homomorphism, if it satisfies the following conditions: Corresponding author addresses: cfx2002@26.com (Feixiang Chen), gleng@staff.shu.edu.cn (Gangsong Leng) Received

2 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), (a) Φ is continuous with respect to the Hausdorff metric. (b) For all K, K 2 K n, Φ(K #K 2 ) = ΦK + ΦK 2, where K #K 2 denotes Blaschke addition (see [9]) of K and K 2, and ΦK + ΦK 2 is the Minkowski addition of ΦK and ΦK 2. (c) For all K K n and every υ SO(n), where SO(n) is the group of rotations of R n. Φ(υK) = υφk, Let S n denote the space of all star bodies in R n endowed with the radial metric. A map Ψ : S n S n is called a radial Blaschke-Minkowski homomorphism if it satisfies the following conditions: (a ) Ψ is continuous with respect to the radial metric. (b ) For all L, L 2 S n, Ψ(L #L2 ) = ΨL +ΨL 2, where L #L2 denotes the radial Blaschke addition (see [8]) of L and L 2, and ΨL +ΨL 2 is the radial Minkowski addition of ΨL and ΨL 2. (c ) For all L S n and every υ SO(n), Ψ(υL) = υψl. Volume inequalities for convex body and star body valued valuations are an active field of research (see [2 4, 7, 9 2, 23, 25]). In the recent paper [22], Wang introduced the following concept of the L p Blaschke-Minkowski homomorphisms: A map Φ p : Ks n Ks n is called an L p Blaschke-Minkowski homomorphism, if it satisfies the following conditions: () Φ p is continuous with respect to the Hausdorff metric. (2) For all K, K 2 K n s, Φ p (K # p K 2 ) = Φ p K + p Φ p K 2, where K # p K 2 denotes L p Blaschke addition of K and K 2, and Φ p K + p Φ p K 2 is the L p Minkowski addition of Φ p K and Φ p K 2. (3) For all K K n s and every υ SO(n), where SO(n) is the group of rotations of R n. Φ p (υk) = υφ p K, In the paper [24], Wang et al. defined L p radial Minkowski homomorphisms as follows: A map Ψ p : S n S n is called an L p radial Minkowski homomorphism, if it satisfies the following conditions: ( ) Ψ p is continuous with respect to the radial metric. (2 ) For all L, L 2 S n, Ψ p (L + n p L 2 ) = Ψ p L + p Ψ p L 2, where L + n p L 2 denotes the radial addition of L and L 2, and Ψ p L + p Ψ p L 2 is the radial Minkowski addition (see [8]) of Ψ p L and Ψ p L 2.

3 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), (3 ) For all L S n and every υ SO(n), Ψ p (υl) = υψ p L. In [9], Schuster has established the following Brunn-Minkowski type inequalities. Theorem. ([9]). Let Φ : K n K n be a Blaschke-Minkowski homomorphism. If K, K 2 K n, then V (Φ(K + K 2 )) V (ΦK ) + V (ΦK 2 ) with equality, if and only if K and K 2 are homothetic. The operator Φ is called even, if ΦK = Φ( K) for all K K n., Theorem.2 ([9]). Let Φ : K n K n be an even Blaschke-Minkowski homomorphism. If K, K 2 K n, then V (Φ (K + K 2 )) V (Φ K ) + V (Φ K 2 ), with equality, if and only if K and K 2 are homothetic. Here Φ K is the polar body of ΦK. The aim of this paper is to establish Brunn-Minkowski type inequalities for L p Blaschke-Minkowski homomorphisms and L p radial Minkowski homomorphisms. Theorem.3. Let Φ p : K n s K n s be an L p Blaschke-Minkowski homomorphism. If K, K 2 K n s and n p, then V (Φ p (K # p K 2 )) p/n V (Φ p K ) p/n + V (Φ p K 2 ) p/n, (.) with equality in (.), if and only if Φ p K and Φ p K 2 are dilates. Theorem.4. Let Φ p : K n s K n s be an L p Blaschke-Minkowski homomorphism. If K, K 2 K n s and n p, then V (Φ p(k # p K 2 )) p/n V (Φ pk ) p/n + V (Φ pk 2 ) p/n, (.2) with equality in (.2), if and only if Φ p K and Φ p K 2 are dilates. Theorem.5. Let Ψ p : S n S n be an L p radial Minkowski homomorphism. If K, K 2 S0 n and 0 < p < n, then V (Ψ p (K + n p K 2 )) p/n V (Ψ p K ) p/n + V (Ψ p K 2 ) p/n, (.3) with equality in (.3), if and only if Ψ p K and Ψ p K 2 are dilates. If p < 0 or p > n, then we get V (Ψ p (K + n p K 2 )) p/n V (Ψ p K ) p/n + V (Ψ p K 2 ) p/n, (.4) with equality (.4), if and only if Ψ p K and Ψ p K 2 are dilates. 2. Notation and background material Let K n denote the set of all convex bodies (compact, convex subsets with non-empty interiors) in R n, and let K n 0 denote the set of convex bodies that contain the origin in their interiors. The subset of Kn 0 consisting of the centered convex bodies will be denoted by K n s. S n is the unit sphere. A convex body is uniquely determined by its support function. The support function of K K n, h(k, ), is defined on S n by h(k, u) = max{u x : x K}. Let δ denote the Hausdorff metric on K n, i.e., for K, L K n, δ(k, L) = h K h L, where denotes the sup-norm on the space of continuous functions, C(S n ).

4 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), Associated with a compact subset L R n, which is star-shaped with respect to the origin, is its radial function ρ(l, ) : S n R, defined by ρ(l, u) = max{λ 0 : λu L}. If ρ(l, ) is positive and continuous, we call L a star body. Let S n and S0 n denote the set of star bodies and the set of star bodies (about the origin) in R n, respectively. Two star bodies K, L are said to be dilates (of one another), if ρ K (u) ρ L (u) is independent of u S n. If K K0 n, then the polar body of K, K, is defined by From (2.), it follows that (K ) = K and K := {x R n : x y, y K}. (2.) h K = ρ K, ρ K = h K. Let K, K 2 K n 0, p, and λ, λ 2 0 (not both 0). The L p Minkowski sum λ K + p λ 2 K 2 is the convex body whose support function is given by (see [5]) h(λ K + p λ 2 K 2, ) p = λ h(k, ) p + λ 2 h(k 2, ) p. For p, the L p -mixed volume V p (K, L) of K, L K n o, can be defined by n p V V (K + p ε L) V (K) p(k, L) = lim. ε 0 + ε In [5], Lutwak has shown that for p, and each K Ko n, there exists a positive Borel measure S p (K, ) on S n, such that the L p -mixed volume V p (K, L) has the following integral representation: V p (K, L) = h p (L, u)ds p (K, u), n S n for all L K n o. The L p -Minkowski inequality states that for K, L K n o and p V p (K, L) V (K) (n p)/n V (L) p/n, with equality, if and only if K and L are dilates. For n p and K, L K n s, the L p -Blaschke addition K + p L K n s was defined in [5] by S p (K# p L, ) = S p (K, ) + S p (L, ). Let K, L S n, and p R and p 0. The L p radial addition K + p ε L is the star body defined by ρ(k + p ε L, ) p = ρ(k, ) p + ερ(l, ) p The L p dual mixed volume V p (K, L) of K, L K n o, can be defined by n p Ṽp(K, V (K + p ε L) V (K) L) = lim. ε 0 + ε The definition above and the polar coordinate formula for volume give the following integral representation of the dual mixed volume Ṽp(K, L) Ṽ p (K, L) = ρ(k, u) n p ρ(l, u) p ds(u). n S n

5 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), Proof of the main results In this section, we give the proofs of our main results Theorem.3.5. First, we need the following lemma. Lemma 3. ([0]). Let K, L S n, if 0 < p < n, then Ṽ p (K, L) V (K) (n p)/n V (L) p/n, with equality, if and only if K and L are dilates. If p < 0 or p > n, then with equality, if and only if K and L are dilates. Ṽ p (K, L) V (K) (n p)/n V (L) p/n, Proof of Theorem.3. Let K, L Ks n and n p. From the definition of L p Blaschke-Minkowski homomorphisms and the L p -Minkowski inequality, for any M K0 n, it follows that V p (M, Φ p (K # p K 2 )) = V p (M, Φ p K + p Φ p K 2 ) = V p (M, Φ p K ) + V p (M, Φ p K 2 ) with equality, if and only if M, Φ p K and Φ p K 2 are dilates. By taking M = Φ p (K # p K 2 ), we get V (M) (n p)/n (V (Φ p K ) p/n + V (Φ p K 2 ) p/n ), V (Φ p (K # p K 2 )) p/n V (Φ p K ) p/n + V (Φ p K 2 ) p/n, with equality, if and only if Φ p K and Φ p K 2 are dilates. Therefore we have proved inequality (.). Proof of Theorem.4. Let K, L Ks n and n p. From the polar coordinate formula for volume and the Minkowski integral inequality, it follows that ( ) p/n V (Φ p(k # p K 2 )) p/n = (h(φ p (K # p K 2 ), u) p ) n/p ds(u) n S n = n p/n h(φ p (K, u)) p + h(φ p (K 2, u)) p n/p n p/n h(φ p (K, u)) p n/p + n p/n h(φ p (K 2, u)) p n/p = V (Φ pk ) p/n + V (Φ pk 2 ) p/n, with equality, if and only if Φ p K and Φ p K 2 are dilates. Therefore we have proved inequality (.2). Proof of Theorem.5. Let K, K 2 S0 n and 0 < p < n. From Lemma 3. and the L p-minkowski inequality, for any M S0 n, it follows that Ṽ p (M, Ψ p (K + n p K 2 )) = Ṽp(M, Ψ p K + p Ψ p K 2 ) = Ṽp(M, Ψ p K ) + Ṽp(M, Ψ p K 2 ) with equality, if and only if M, Ψ p K and Ψ p K 2 are dilates. By taking M = Ψ p (K + n p K 2 ), we get V (M) (n p)/n (V (Ψ p K ) p/n + V (Ψ p K 2 ) p/n ), V (Ψ p (K + n p K 2 )) p/n V (Ψ p K ) p/n + V (Ψ p K 2 ) p/n,

6 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), with equality, if and only if Ψ p K and Ψ p K 2 are dilates. Therefore we have proved inequality (.3). If p < 0 or p > n, then we get V (Ψ p (K + n p K 2 )) p/n V (Ψ p K ) p/n + V (Ψ p K 2 ) p/n, with equality, if and only if Ψ p K and Ψ p K 2 are dilates. The inequality (.4) is proved. Since the L p projection body operator Π p is an L p Blaschke-Minkowski homomorphism, we get the following inequalities which were established by Lu and Leng in []. Corollary 3.2 ([]). Let Π p : K n s K n s be the L p projection body operator. If K, K 2 K n s and n p, then V (Π p (K # p K 2 )) p/n V (Π p K ) p/n + V (Π p K 2 ) p/n, (3.) V (Π p(k # p K 2 )) p/n V (Π pk ) p/n + V (Π pk 2 ) p/n, (3.2) with equality in (3.) and (3.2), if and only if Π p K and Π p K 2 are dilates. Acknowledgment We are grateful to the referee for the suggested improvements. We would like to acknowledge the support from the National Natural Science Foundation of China (67249) and the Natural Science Foundation of Chongqing Municipal Education Commission (No. KJ50009). References [] J. Abardia, A. Bernig, Projection bodies in complex vector spaces, Adv. Math., 227 (20), [2] J. Abardia, T. Wannerer, Aleksandrov-Fenchel inequalities for unitary valuations of degree 2 and 3, Calc. Var. Partial Differential Equations, 54 (205), [3] S. Alesker, A. Bernig, F. E. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal., 2 (20), [4] A. Berg, L. Parapatits, F. E. Schuster, M. Weberndorfer, Log-concavity properties of Minkowski valuations, arxiv, 204 (204), 44 pages. [5] R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.), 39 (2002), [6] R. J. Gardner, Geometric tomography, Second edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York, (2006). [7] R. J. Gardner, P. Gronchi, A Brunn-Minkowski inequality for the integer lattice, Trans. Amer. Math. Soc., 353 (200), [8] R. J. Gardner, D. Hug, W. Weil, Operations between sets in geometry, J. Eur. Math. Soc. (JEMS), 5 (203), [9] R. J. Gardner, L. Parapatits, F. E. Schuster, A characterization of Blaschke addition, Adv. Math., 254 (204), [0] C. Haberl, L p intersection bodies, Adv. Math., 27 (2008), [] F.-H. Lu, G.-S. Leng, On L p-brunn-minkowski type inequalities of convex bodies, Bol. Soc. Mat. Mexicana, 3 (2007), , 3.2 [2] M. Ludwig, Projection bodies and valuations, Adv. Math., 72 (2002), [3] M. Ludwig, Intersection bodies and valuations, Amer. J. Math., 28 (2006), [4] E. Lutwak, Inequalities for mixed projection bodies, Trans. Amer. Math. Soc., 339 (993), [5] E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (993), [6] R. Osserman, The Brunn-Minkowski inequality for multiplictities, Invent. Math., 25 (996), [7] L. Parapatits, F. E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math., 230 (202), [8] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, (993). [9] F. E. Schuster,Volume inequalities and additive maps of convex bodies, Mathematika, 53 (2006), ,.,.2 [20] F. E. Schuster, Valuations and Busemann-Petty type problems, Adv. Math., 29 (2008),

7 F. X. Chen, G. S. Leng, J. Nonlinear Sci. Appl. 9 (206), [2] F. E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J., 54 (200), 30. [22] W. Wang, L p Blaschke-Minkowski homomorphisms, J. Inequal. Appl., 203 (203), 4 pages. [23] W. Wang, L p Brunn-Minkowski type inequalities for Blaschke-Minkowski homomorphisms, Geom. Dedicata, 64 (203), [24] W. Wang, L.-J. Liu, B.-W. He, L p radial Minkowski homomorphisms, Taiwanese J. Math., 5 (20), [25] T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J., 60 (20),