Independence for Sets of Topological Spheres

1991 ◽  
Vol 34 (4) ◽  
pp. 520-524
Author(s):  
Lewis Pakula ◽  
Sol Schwartzman
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Necessary Condition ◽  
Number Of Components

AbstractConsider a collection of topological spheres in Euclidean space whose intersections are essentially topological spheres. We find a bound for the number of components of the complement of their union and discuss conditions for the bound to be achieved. This is used to give a necessary condition for independence of these sets. A related conjecture of Griinbaum on compact convex sets is discussed.

Killed Brownian motion and the Brunn–Minkowski inequalities

2012 ◽  
Vol 153 (1) ◽  
pp. 111-121
Author(s):  
HUILING LE
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Poisson Equation ◽  
Harmonic Functions ◽  
Convex Sets ◽  
Compact Convex ◽  
Compact Sets ◽  
Brownian Motions ◽  
The Poisson Equation ◽  
Killed Brownian Motions

AbstractWe construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

Some extremal properties of convex sets

1975 ◽  
Vol 77 (3) ◽  
pp. 515-524 ◽  
Author(s):  
J. N. Lillington
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Convex Sets ◽  
Compact Convex ◽  
Upper And Lower Bounds ◽  
Arbitrary Convex ◽  
Extremal Properties ◽  
Convex Subsets

In this paper we shall suppose that all convex sets are compact convex subsets of Euclidean space En. We shall be concerned in producing upper and lower bounds for the ‘total edge lengths’ of simplices which are contained in or contain arbitrary convex sets in terms of the inradii and circumradii of these sets. However, before proceeding further, we shall introduce some notation and give some motivation for this work.

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.

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 Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

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