Convex analysis in groups and semigroups: a sampler

TL;DR

This paper extends convex analysis to monoids, showing that many classical results hold in this broader algebraic setting, supported by examples and counterexamples.

Contribution

It introduces a canonical definition of convexity in monoids and demonstrates the applicability of convex analysis results beyond vector spaces.

Findings

Classical convex analysis results are valid in monoids.

Examples illustrate the applicability of convexity in groups and semigroups.

Counter-examples highlight limitations of the approach.

Abstract

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and counter-examples are also discussed.