On the boundary of closed convex sets in $E^{n}$

TL;DR

This paper generalizes the Krein-Milman theorem to certain non-compact convex sets in Euclidean space, providing conditions for convexity, affinity, and starshapedness based on boundary points.

Contribution

It introduces a class of closed convex sets as convex hulls of their profiles and establishes new necessary and sufficient boundary conditions.

Findings

Characterization of convex sets as convex hulls of profiles

Generalization of Krein-Milman theorem to non-compact sets

Boundary conditions for convexity, affinity, and starshapedness

Abstract

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starshapedness of a closed set and its boundary have been derived in terms of their boundary points.