# Extension Of The Bauer's Maximum Principle For Compact Metrizable Sets

## TL;DR

This paper extends Bauer's maximum principle to compact metrizable convex sets, showing that functions attaining their maximum at a unique extremal point form a dense Gδ subset in the space of convex upper semi-continuous functions.

## Contribution

It generalizes Bauer's maximum principle to compact metrizable sets and characterizes the generic functions attaining their maximum uniquely at an extremal point.

## Key findings

- The set of functions attaining maximum at exactly one extremal point is dense.
- This set is a Gδ subset in the space of convex upper semi-continuous functions.
- The extension applies to compact metrizable convex sets.

## Abstract

Let X be a nonempty convex compact subset of some Haus-dorff locally convex topological vector space S. The well know Bauer's maximum principle stats that every convex upper semi-continuous function from X into R attains its maximum at some extremal point of X. We give some extensions of this result when X is assumed to be compact metrizable. We prove that the set of all convex upper semi-continuous functions attaining there maximum at exactly one extremal point of X is a G $\delta$ dense subset of the space of all convex upper semi-continuous functions equipped with a metric compatible with the uniform convergence .