M-infinitely divisible random compact convex sets

1985 ◽  
pp. 226-248 ◽  
Author(s):  
Evarist Giné ◽  
Marjorie G. Hahn
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Infinitely Divisible

Random compact convex sets which are infinitely divisible with respect to Minkowski addition

1979 ◽  
Vol 11 (4) ◽  
pp. 834-850 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Infinite Divisibility ◽  
Spectral Measures ◽  
Infinitely Divisible ◽  
Minkowski Addition ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Canonical Representations ◽  
Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

Random compact convex sets which are infinitely divisible with respect to Minkowski addition

1979 ◽  
Vol 11 (04) ◽  
pp. 834-850 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Infinite Divisibility ◽  
Spectral Measures ◽  
Infinitely Divisible ◽  
Minkowski Addition ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Canonical Representations ◽  
Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
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.

Facial structure of the sum of two compact convex sets

1972 ◽  
Vol 197 (3) ◽  
pp. 189-196 ◽  
Author(s):  
A. K. Roy
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Facial Structure

SYMMETRIES OF COMPACT CONVEX SETS

1974 ◽  
Vol 25 (1) ◽  
pp. 323-328 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Convex Sets ◽  
Compact Convex

scholarly journals A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of Rn

2015 ◽  
Vol 194 ◽  
pp. 125-143
Author(s):  
Bernardo González Merino ◽  
Natalia Jonard-Pérez
Keyword(s):  
Convex Sets ◽  
Compact Convex

scholarly journals On the number of minimal pairs of compact convex sets that are not translates of one another

2003 ◽  
Vol 158 (1) ◽  
pp. 59-63 ◽  
Author(s):  
J. Grzybowski ◽  
R. Urbański
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Minimal Pairs

scholarly journals Extremal Minkowski additive selections of compact convex sets

1989 ◽  
Vol 105 (3) ◽  
pp. 697-697
Author(s):  
Rade T. Živaljevi{ć
Keyword(s):  
Convex Sets ◽  
Compact Convex
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