Characterizations of Faces of Convex Sets in Infinite-dimensional Vector Spaces

TL;DR

This paper presents three equivalent characterizations of faces in convex sets within infinite-dimensional real vector spaces, extending finite-dimensional results through generalized semispaces, preorders, and step-affine functions.

Contribution

It introduces three novel, equivalent characterizations of faces in infinite-dimensional convex sets, generalizing finite-dimensional lexicographical characterizations.

Findings

All three characterizations are equivalent.

The characterizations extend finite-dimensional results.

They apply to infinite-dimensional real vector spaces.

Abstract

In the paper three different characterizations of faces of convex sets, belonging to infinite-dimensional real vector spaces, are presented. The first one is formulated in the terms of generalized semispaces, the second -- in the terms of compatible complete (total) preorders, and the third -- in the terms of step-affine functions. All three characterization are equivalent each other and extend to infinite-dimensional vector spaces the lexicographical characterization of faces established in finite-dimensional settings by Martinez-Legaz J.-E. (Acta Mathematica Vietnamica. 1997. Vol. 22, No.~1, P. 207--211).