Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J. Math. Anal. 2 (2008), no., Banach Journal of Mathematical Analysis ISSN: (electronic) WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES WING-SUM CHEUNG AND CHANG-JIAN ZHAO 2 Submitted by Th. M. Rassias Abstract. The main purposes of this paper are to establish some new Brunn Minkowski inequalities for width-integrals of mixed projection bodies and affine surface area of mixed bodies, together with their inverse forms.. Introduction In recent years some authors including Ball [], Bourgain [2], Gardner [3], Schneider [4] and Lutwak [5] [0] et al have given considerable attention to Brunn Minkowski theory, Brunn Minkowski-Firey theory, and their various generalizations. In particular, Lutwak [7] generalized the Brunn Minkowski inequality (.) to mixed projection bodies and obtain inequality (.2): The Brunn Minkowski inequality If K, L K n, then V (K + L) /n V (K) /n + V (L) /n, with equality if and only if K and L are homothetic. Date: Received: 9 April 2008; Accepted: 5 May Corresponding author. Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project No.: HKU706/07P). Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China ( ) Mathematics Subject Classification. 52A40. Key words and phrases. Width-integrals, Affine surface area, Mixed projection body, Mixed body. 70

2 WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES 7 The Brunn Minkowski inequality for mixed projection bodies If K, L K n, then V (Π(K + L)) /n(n ) V (ΠK) /n(n ) + V (ΠL) /n(n ), with equality if and only if K and L are homothetic. On the other hand, width-integrals of convex bodies and affine surface areas play an important role in Brunn Minkowski theory. Width-integrals were first considered by Blaschke [] and later by Hadwiger [2]. In addition, Lutwak established the following results for the width-integrals of convex bodies and affine surface areas. The Brunn Minkowski inequality for width-integrals of convex bodies [0] If K, L K n, i < n, B i (K + L) /(n i) B i (K) /(n i) + B i (L) /(n i), (.) with equality if and only if K and L have similar width. The Brunn Minkowski inequality for affine surface area [9] If K, L K n, and i R, then for i <, Ω i (K +L) (n+)/(n i) Ω i (K) (n+)/(n i) + Ω i (L) (n+)/(n i), (.2) with equality if and only if K and L are homothetic, while for i >, Ω i (K +L) (n+)/(n i) Ω i (K) (n+)/(n i) + Ω i (L) (n+)/(n i), (.3) with equality if and only if K and L are homothetic. The purpose of this paper is two-fold. First, to generalize inequality (.) to the context of to mixed projection bodies and also establish its inverse version. Second, to obtain analogs of inequalities (.2) and (.3) for affine surface area of mixed bodies. 2. Notations and Preliminary works The setting for this paper is the 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 non-empty interiors) in R n. For p K n, let K n p denote the subset of K n that contains the centered (centrally symmetric with respect to p) bodies. We reserve the letter u for unit vectors, and the letter B is reserved for the unit ball centered at the origin. The boundary surface of B is S n. For u S n, 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. 2.. Mixed volumes. Let K K n. We denote by V (K) the n-dimensional volume of K. Let h(k, ) : S n R denote the support function of K, i.e., h(k, u) := Max{u x : x K}, u S n, where u x denotes the usual inner product of u and x in R n.

3 72 W.S. CHEUNG, C. ZHAO 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 ). For any nonnegative scalar λ, we write λk := {λx : x K}. For K i K n, λ i 0, (i =, 2,..., r), the Minkowski linear combination r i= λ ik i K n is defined by λ K + + λ r K r := {λ x + + λ r x r K n : x i K i }. It is trivial to verify that h(λ K + + λ r K r, ) = λ h(k, ) + + λ r h(k r, ). (2.) If K i K n and λ i (i =, 2,..., r) are nonnegative real numbers, then of fundamental importance is the fact that the volume of r i= λ ik i is a homogeneous polynomial in λ i given by [4] V (λ K + + λ r K r ) = λ i λ in V i...i n, (2.2) i,...,i n where the sum is taken over all n-tuples (i,..., i n ) of positive integers not exceeding r. The coefficient V i...i n depends only on the bodies K i,..., K in, and is uniquely determined by (2.2). It is called the mixed volume of K i,..., K in, and is also written as V (K i,..., K in ). If K i = = K n i = K and K n i+ = = K n = L, then the mixed volume V (K... K n ) is usually written as V i (K, L). If L = B, then V i (K, B) is the ith projection measure(quermassintegral) of K and is written as W i (K). With this notation, W 0 = V (K), while nw (K) is the surface area S(K) of K Width-integrals of convex bodies. For u S n, b(k, u) is defined to be half the width of K in the direction u. Two convex bodies K and L are said to have similar width if there exists a constant λ > 0 such that b(k, u) = λb(l, u) for all u S n. For K K n and p intk, we use K p to denote the polar reciprocal of K with respect to the unit sphere centered at p. The width-integral of index i is defined by Lutwak [0]: For K K n and i R, B i (K) := b(k, u) n i ds(u), (2.3) n S n where ds is the (n )-dimensional volume element on S n. The width-integral of index i is hence a map B i : K n R which is positive, continuous, homogeneous of degree n i and invariant under rigid motions. In addition, for i n, it is also bounded and monotone under set inclusion. The following results (cf. [0]) will be used later: b(k + L, u) = b(k, u) + b(l, u), (2.4) B 2n (K) V (K p ), (2.5)

4 WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES 73 with equality if and only if K is symmetric with respect to p The radial function and the Blaschke linear combination. The radial function of the convex body K is the function ρ(k, ) : S n R defined for u S n by ρ(k, ) := Max{λ 0 : λµ K}. If ρ(k, ) is positive and continuous, K will be called a star body. Let ϕ n denote the set of star bodies in R n. A convex body K is said to have a positive continuous curvature function [5] f(k, ) : S n [0, ), if for each L ϕ n, the mixed volume V (K, L) has the integral representation V (K, L) = f(k, u)h(l, u)ds(u). n S n The subset of K n consisting of convex bodies which have a positive continuous curvature function will be denoted by κ n. Let κ n c denote the set of centrally symmetric members of κ n. For K κ n, it is shown in [6] that uf(k, u)ds(u) = 0. S n Suppose K, L κ n and λ, µ 0 be not both zero. From above it follows that the function λf(k, ) + µf(l, ) satisfies the hypothesis of Minkowski s existence theorem (see [3]). The solution of the Minkowski problem for this function is denoted by λ K +µ L, that is, f(λ K +µ L, ) = λf(k, ) + µf(l, ), (2.6) where the linear combination λ K +µ L is called a Blaschke linear combination. The relationship between Blaschke and Minkowski scalar multiplication is given by λ K = λ /(n ) K Mixed affine area and mixed bodies. The affine surface area Ω(K) of K κ n is defined by Ω(K) := f(k, u) n/(n+) ds(u). S n It is well known that this functional is invariant under unimodular affine transformations. For K, L κ n and i R, the ith mixed affine surface area Ω i (K, L) of K and L was defined in [5] by Ω i (K, L) := f(k, u) (n i)/(n+) f(l, u) i/(n+) ds(u). S n Now, we define the ith affine area Ω i (K) of K κ n to be Ω i (K, B). Since f(b, ) =, one has Ω i (K) = f(k, u) (n i)/(n+) ds(u), i R. S n (2.7)

5 74 W.S. CHEUNG, C. ZHAO Lutwak [8] defined mixed bodies of convex bodies K,..., K n as [K,..., K n ]. The following property will be used later: [K + K 2, K 3,..., K n ] = [K, K 3,..., K n ] +[K 2, K 3,..., K n ]. (2.8) 2.5. Mixed projection bodies and their polars. If K is a convex body that contains the origin in its interior, we define the polar body K of K by K := {x R n x y, y K}. If K i (i =, 2,..., n ) K n, then the mixed projection body of K i (i =, 2,..., n ) is denoted by Π(K,..., K n ), and whose support function is given, for u S n, by [7] h (Π(K,..., K n ), u) := v(k u,..., K u n ). It is easy to see that Π(K,..., K n ) is centered. We use Π (K,..., K n ) to denote the polar body of Π(K,..., K n ). It is also called the polar of mixed projection body of K i (i =, 2,..., n ). If K = = K n i = K and K n i = = K n = L, then Π(K,..., K n ) will be written as Π i (K, L). If L = B, then Π i (K, B) is called the ith projection body of K and is denoted by Π i K. We write Π 0 K as ΠK. We will simply write Π i K and Π K rather than (Π i K) and (ΠK), respectively. The following property will be used: Π(K 3,..., K n, K + K 2 ) = Π(K 3,..., K n, K ) + Π(K 3,..., K n, K 2 ). (2.9) 3. Main results Our main results are the following Theorems. Theorem 3.. Let K, K 2,..., K n K n and C = (K 3,..., K n ). (i) For i < n, B i (Π(C, K + K 2 )) /(n i) B i (Π(C, K )) /(n i) + B i (Π(C, K 2 ) /(n i), (3.) with equality if and only if Π(C, K ) and Π(C, K 2 ) are homothetic. (ii) For i > n, B i (Π(C, K + K 2 )) /(n i) B i (Π(C, K )) /(n i) + B i (Π(C, K 2 ) /(n i), (3.2) with equality if and only if Π(C, K ) and Π(C, K 2 ) are homothetic. Proof. Here, we only give a proof of (ii).

6 WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES 75 From (2.), (2.3), (2.4), (2.9) and applying the Minkowski inequality for integrals [4, p. 47], we obtain B i (Π(C, K + K 2 )) /(n i) ) /(n i) = b(π(c, K + K 2 ), u) n i ds(u) n S n = b(π(c, K ) + Π(C, K 2 ), u) n i ds(u) n S n = (b(π(c, K ), u) + b(π(c, K 2 ), u)) n i ds(u) n S n ) /(n i) b(π(c, K ), u) n i ds(u) n S n ) /(n i) + b(π(c, K ), u) n i ds(u) n S n = B i (Π(C, K )) /(n i) + B i (Π(C, K 2 )) /(n i), ) /(n i) ) /(n i) with equality if and only if Π(C, K ) and Π(C, K 2 ) have similar width. In view of Π(C, K ) and Π(C, K 2 ) are centered (centrally symmetric with respect to origin), we conclude that the equality holds if and only if Π(C, K ) and Π(C, K 2 ) are homothetic. Taking i = 0, inequality (3.) reduces to the following result: Corollary 3.2. If K, K 2,..., K n K n, and C = (K 3,..., K n ), then B(Π(C, K + K 2 )) /n B(Π(C, K )) /n + B(Π(C, K 2 ) /n, with equality if and only if Π(C, K ) and Π(C, K 2 ) are homothetic. Taking i = 2n, inequality (3.2) reduces to the following result: Corollary 3.3. If K, K 2,..., K n K n, and C = (K 3,..., K n ), then B 2n (Π(C, K + K 2 )) /n B 2n (Π(C, K )) /n + B 2n (Π(C, K 2 ) /n, (3.3) with equality if and only if Π(C, K ) and Π(C, K 2 ) are homothetic. From (2.5), (3.3), and noting that the projection body is centered(centrally symmetric with respect to origin), we obtain the following Brunn Minkowski inequality of polars of mixed projection bodies. Corollary 3.4. If K, K 2,..., K n K n and C = (K 3,..., K n ), then V (Π (C, K + K 2 )) /n V (Π (C, K ) /n + V (Π (C, K 2 )) /n, with equality if and only if Π(C, K ) and Π(C, K 2 ) are homothetic.

7 76 W.S. CHEUNG, C. ZHAO Theorem 3.5. Suppose K, K 2,..., K n K n and all mixed bodies of K,..., K n have positive continuous curvature functions. (i) For i <, Ω i ([K + K 2, K 3,..., K n ]) (n+)/(n i) Ω i ([K, K 3, K 4..., K n ]) (n+)/(n i) + Ω i ([K 2, K 3,..., K n ]) (n+)/(n i), with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3..., K n ] are homothetic. (ii) For i >, Ω i ([K + K 2, K 3,..., K n ]) (n+)/(n i) Ω i ([K, K 3, K 4,..., K n ]) (n+)/(n i) + Ω i ([K 2, K 3,..., K n ]) (n+)/(n i), with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3..., K n ] are homothetic. Proof. (i) From (2.6), (2.7), (2.8) and in view of the Minkowski inequality for integrals [4, p. 47], we obtain Ω i ([K + K 2, K 3, K 4,..., K n ]) (n+)/(n i) ) (n+)/(n i) = f([k + K 2, K 3, K 4,..., K n ], u) (n i)/(n+) ds(u) S n = f([k, K 3,..., K n ] +[K 2, K 3,..., K n ], u) (n i)/(n+) ds(u) S n = (f([k,..., K n ], u) + f([k 2,..., K n ], u)) (n i)/(n+) ds(u) S n ) (n+)/(n i) f([k, K 3, K 4,..., K n ], u) (n i)/(n+) ds(u) S n ) (n+)/(n i) + f([k 2, K 3,..., K n ], u) (n i)/(n+) ds(u) S n = Ω i ([K, K 3, K 4,..., K n ]) (n+)/(n i) + Ω i ([K 2, K 3,..., K n ]) (n+)/(n i), ) (n+)/(n i) ) (n+)/(n i) with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3,..., K n ] are homothetic. (ii) Similarly, from (2.6), (2.7), (2.8) and in view of inverse Minkowski inequality [4, p. 47], we can obtain (3.5). Taking i = 0 in (3.5), we have Corollary 3.6. If K, K 2,..., K n K n and all mixed bodies of K, K 2,..., K n have positive continuous curvature functions, then Ω([K + K 2, K 3,..., K n ]) (n+)/n Ω([K, K 3, K 4,..., K n ]) (n+)/n + Ω([K 2, K 3,..., K n ]) (n+)/n, with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3..., K n ] are homothetic.

8 WIDTH-INTEGRALS AND AFFINE SURFACE AREA OF CONVEX BODIES 77 Taking i = 2n in (3.5), we have Corollary 3.7. If K, K 2,..., K n K n and all mixed bodies of K, K 2,..., K n have positive continuous curvature functions, then Ω 2n ([K + K 2, K 3,..., K n ]) (n+)/n Ω 2n ([K, K 3, K 4..., K n ]) (n+)/n + Ω 2n ([K 2, K 3,..., K n ]) (n+)/n, with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3..., K n ] are homothetic. Taking i = n in (3.5), we have Corollary 3.8. If K, K 2,..., K n K n and all mixed bodies of K, K 2,..., K n have positive continuous curvature functions, then Ω n ([K + K 2, K 3,..., K n ]) (n+)/2n Ω n ([K, K 3, K 4..., K n ]) (n+)/2n + Ω n ([K 2, K 3,..., K n ]) (n+)/2n, with equality if and only if [K, K 3, K 4,..., K n ] and [K 2, K 3..., K n ] are homothetic. References. K. Ball, Volumes of sections of cubes and related problems, Geometric aspects of functional analysis (987 88), , Lecture Notes in Math., 376, Springer, Berlin, J. Bourgain and J. Lindenstrauss, Projection bodies, Geometric aspects of functional analysis (986/87), , Lecture Notes in Math., 37, Springer, Berlin, R.J. Gardner, Geometric Tomography, Cambridge University Press, Cambridge, R. Schneider, Convex bodies: The Brunn Minkowski Theory, Cambridge University Press, Cambridge, E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60 (990), E. Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (985), E. Lutwak, Inequalities for mixed projection, Tans. Amer. Math. Soc. 339 (993), E. Lutwak, Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (986), E. Lutwak, Mixed affine surface area, J. Math. Anal. Appl. 25 (987), E. Lutwak, Width-integrals of convex bodies, Proc. Amer. Math. Soc. 53 (975), W. Blaschke, Vorlesungen über Integral geometric I, II, Teubner, Leipzig, 936, 937; reprint, chelsea, New York, H. Hadwiger, Vorlesungen über inhalt, Oberfläche und isoperimetrie, Springer, Berlin, T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer, Berlin, G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press. Cambridge, 934. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. address: wscheung@hku.hk 2 Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 3008, P.R.China. address: chjzhao@yahoo.com.cn