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Some inequalities about mixed volumes

Abstract : We prove inequalities about the quermassintegrals V-k(K) of a convex body K in R-n (here, V-k(K) is the mixed volume V ((K, k), (B-n, n - k)) where B-n is the Euclidean unit ball). (i) The inequality V-k(K + L)/Vk-1(K + L) greater than or equal to V-k(K)/Vk-1(K) + V-k(L)/Vk-1(L) holds for every pair of convex bodies K and L in R-n if and only if k = 2 or k = 1. (ii) Let 0 less than or equal to k less than or equal to p less than or equal to n. Then, for every p-dimensional subspace E of R-n, Vn-k(K)/\K\ greater than or equal to 1/((n-p) (n-p+k)) Vp-k(PEK)/\PEK\ where PEK denotes the orthogonal projection of K onto E. The proof is based on a sharp upper estimate for the volume ratio \K\/\L\ in terms of Vn-k(K)/Vn-k(L), whenever L and K are two convex bodies in R-n such that K subset of or equal to L.
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Submitted on : Wednesday, May 2, 2012 - 6:35:08 PM
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Matthieu Fradelizi, A Giannopoulos, Mathieu Meyer. Some inequalities about mixed volumes. Israël Journal of Mathematics, Hebrew University Magnes Press, 2003, 135 (?), pp.157--179. ⟨10.1007/BF02776055⟩. ⟨hal-00693589⟩



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