Functionals related to mixed volumes
نویسندگان
چکیده
منابع مشابه
Computing Mixed Discriminants , Mixed Volumes
We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite n × n matrices within a 2O(n) factor. As a corollary, we show that the permanent of an n×n nonnegative matrix and the mixed volume of n ellipsoids inRn can be computed within a 2O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximat...
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In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called “mixed Eulerian numbers”, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially...
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In this paper we consider the following analog of Bezout inequality for mixed volumes: V (P1, . . . , Pr,∆ )Vn(∆) r−1 ≤ r ∏ i=1 V (Pi,∆ ) for 2 ≤ r ≤ n. We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional...
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We consider functionals measuring the dispersion of a d-dimensional distribution which are based on the volumes of simplices of dimension k ≤ d formed by k + 1 independent copies and raised to some power δ. We study properties of extremal measures that maximize these functionals. In particular, for positive δ we characterize their support and for negative δ we establish connection with potentia...
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We give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Eulerian numbers. 1998 Academic Press Let C d denote the d-dimensional unit cube, that is, C d =[x # R d : 0x i 1]. All the volumes and mixed volumes which appear in this note will be normalized so that the volume of C d is given by V(C d)=d!. We consider slices...
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ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 1961
ISSN: 0019-2082
DOI: 10.1215/ijm/1255630888