Centre of a compact convex set

TL;DR

This paper introduces the concept of the centre of a convex set, explores the structure of affine function spaces on such sets, and characterizes related base normed spaces and their properties.

Contribution

It defines the centre of a convex set and characterizes the dual spaces of base normed spaces with a centre, providing new insights into their structure.

Findings

Spaces of continuous affine functions are dual to base normed spaces with a centre.

Characterization of when these function spaces are absolute order unit spaces.

Conditions for base normed spaces to be absolutely base normed spaces.

Abstract

We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space so that the latter becomes an absolutely base normed space.