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- W2564261685 abstract "One of the major outstanding questions in Geometric Convexity is Petty's conjectured inequality between the volume of a convex body and that of its projection body. It is shown that if Petty's conjectured inequality holds, then it is the first of a family of such inequalities (involving mixed projection bodies). All of the members of this family are strengthened versions of the classical inequalities between pairs of Quermassintegrals of a convex body. The last member of this family (of conjectured inequalities) is established. Let denote the space of convex figures (compact, convex subsets) of Euclidean n-space, (n > 2), endowed with the topology induced by the Hausdorff metric. The dimension of a convex figure is the dimension of a minimal dimensional fiat (affine subspace) which contains the figure. Let )f~ denote the subset of ~¢ consisting of figures whose dimension is at least n - i. Thus, ~ denotes the space of convex bodies (convex figures with non-empty interiors) in R. For • let h(K,. ): S- 1 ~ R denote the (restricted) support function of K. For the ith n-dimensional Quermassintegral (projection measure) of we write W~(K). Thus, Wo(K) is the volume of K, nWI(K) is the surface area of K, while 2W,_ l(K)/co, is the mean width of K, where co, is the n-dimensional volume of the unit ball, B, in R. For a unit vector uS- 1, let K denote the image of the orthogonal projection of onto ~,, the codimension-1 subspace of orthogonal to u. For the ith (n - 1)-dimensional Quermassintegral of the figure K in ~, we write wi(K). For ~ Jr, the ith projection body of K, IIiK, is the centrally symmetric convex figure whose support function is given by (1) h(HiK, u) = wi(K) for uS- 1. (Note that the indexing of mixed projection bodies follows (5), (6), (10) and differs from that used in the survey of Schneider and Weil (14).) Thus, for 2 > O, 1-Ii2K = 2-i- 'IIiK. For the second ith projection body of K, 1-I~(l-IiK), write H2K." @default.
- W2564261685 created "2017-01-06" @default.
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- W2564261685 date "1990-01-01" @default.
- W2564261685 modified "2023-09-24" @default.
- W2564261685 title "Dedicated to Prof. Dr Ludwig Danzer on the occasion of his 60th birthday" @default.
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