A characterization of unbounded Choquet simplices

Abstract

In this paper, we give a property of the linearly dosed non-linearly bounded Choquet simplices of a real vector space which can be used to describe the dosed Choquet simplices of E ". Such a description can be obtained from a theorem of Gruber [3, Satz 2] using other techniques.1. The aim of this paper is to provide a characterization of those unbounded lcosed convex subsets S of E" which are Choquet simplices. This will answer a question of Eggleston ([2], I, p. 210) and complete [7] and [9]. Recall that a Choquet simplex (briefly, a simplex) of a real vector space E is a non-empty convex subset S of E such that: ~Sn(S+x)= either [,~S + y, ;~ >~ O, y ~ E. 2. Let us first consider a real vector space E (not necessarily finite-dimensional). Remember ([5], [6]) that a convex subset of E is linearly closed (resp. linearly bounded) if its intersection with every straight line in E is dosed (resp. bounded) for the natural topology of the line. Following Kendall ([6]) we say that a convex linearly dosed and linearly bounded subset of E is linearly compact.To end with we define an algebraically exposed point of a convex subset C of E as a point x ~ C such that there is a closed half-space a"eg for which ~n C = (x}.It is well-known or clear that, if E is finite-dimensional, those notions coincide with the corresponding topological notions for the natural topology of E.3. The following theorem gives a property of a certain kind of simplices in E. THEOREM 1. A linearly closed non-linearly bounded simplex S has at most one algebraically exposed point.Suppose that S has two different algebraically exposed points. One can assume that one of these points is 0; the other one will be denoted by e.There are closed half-spaces 9fro, whose bounding hyperplane Ho contains 0, and ~, whose bounding hyperplane H~ contains e, such that ~0nS={0} and ~enS={e}.Geometriae Dedicata 6 (1977) 95-98. All Rights Reserved

The Characteristic Intersection Property of Line-Free Choquet Simplices in E d

“…Thus z corresponds to the exposed point u of B 1 . (1) of Theorem 2, if B is bounded and B 2 is unbounded, then for any vector v ∈ E d the set int B 1 contains at most one exposed point of the body v + B 2 .…”

“…Similarly, B 2 is a translate of B. Summing up, we get that B satisfies equation (1), i.e., B is a Choquet simplex. From [1] it follows that B is a simplicial d-cone.…”

“…From [1] and [6] it follows that a line-free convex body B ⊂ E d is a Choquet simplex if and only if B is either a usual d-simplex of the form conv(x 0 , x 1 , . .…”

A Characterization of Homothetic Simplices

“…Independently, this assertion was strengthened by Eggleston [7] who showed, confirming a conjecture of Gruber [12], that a bounded measurable set of positive measure in E d satisfying condition (2) is the interior of a d-simplex together with the relative interiors of some of its faces. Bair and Fourneau [2] proved that a closed, unbounded, and line-free Choquet simplex in E d is a convex cone whose base is a k-simplex (k ≤ d). Fourneau [9]- [11] studied nonclosed unbounded Choquet simplices in E d (see also [1] and [3]).…”