REMARKS ABOUT MIXED DISCRIMINANTS AND VOLUMES

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in ℝn. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

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Diameters in Typical Convex Bodies

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-003-8 ◽

1990 ◽

Vol 42 (1) ◽

pp. 50-61 ◽

Cited By ~ 7

Author(s):

Imre Bárány ◽

Tudor Zamfirescu

Keyword(s):

Convex Body ◽

Convex Sets ◽

Compact Convex ◽

Hausdorff Metric ◽

Convex Bodies ◽

General Setting ◽

Baire Space ◽

Supporting Hyperplanes ◽

The World

The most usual diameters in the world are those of a sphere and they all contain its centre. More generally, a chord of a convex body in Rd is called a diameter if there are two parallel supporting hyperplanes at the two endpoints of the chord.It is easily seen that there are points on at least two diameters. From a result of Kosiński [6] proved in a more general setting it follows that every convex body has a point lying on at least three diameters. Does a typical convex body behave like a sphere and contain a point on infinitely or even uncountably many diameters?But what is a typical convex body? The space 𝒦 of all convex bodies (d-dimensional compact convex sets) in Rd, equipped with the Hausdorff metric, is a Baire space.

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Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

Algebra and Logic ◽

10.1007/s10469-009-9065-x ◽

2009 ◽

Vol 48 (5) ◽

pp. 330-343 ◽

Cited By ~ 5

Author(s):

K. Keimel

Keyword(s):

Convex Sets ◽

Compact Convex ◽

Compact Spaces

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An elementary renewal theorem for random compact convex sets

Advances in Applied Probability ◽

10.2307/1427929 ◽

1995 ◽

Vol 27 (4) ◽

pp. 931-942 ◽

Cited By ~ 4

Author(s):

Ilya S. Molchanov ◽

Edward Omey ◽

Eugene Kozarovitzky

Keyword(s):

Support Function ◽

Convex Sets ◽

Compact Convex ◽

Renewal Function ◽

Renewal Theorem ◽

Closed Sets ◽

Random Closed Sets ◽

Minkowski Sums ◽

Image Position ◽

Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

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