A Property of Extreme Points of Compact Convex Sets

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

Extreme Points of Polar Convex Sets

Approximating Extreme Points of Infinite Dimensional Convex Sets

The property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in IRn is attained at an extreme point is generalized by the Bauer Minimum Principle to the infinite dimensional context. The problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem. Consider now an infinite dimensio...

Extreme Points in Convex Sets of Symmetric Matrices

This paper deals with the following problem: What are the extreme points of a convex set K of n X n matrices, which is the intersection of the set S„ of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices HI In the case where the facial structure of H is known, we expose a general method to determine the extreme points of K (Theorem 1). Then, we apply this m...

A Combinatorial Property of Convex Sets

A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn , c a constant. We prove: Let 8 be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements Si , Sj of 8 such that any set S′ homothetic to S that con...