Barycentric algebras -- convexity and order

TL;DR

This paper explores the algebraic structure of barycentric algebras, examining their properties, examples, and applications in geometry and physics, highlighting their relevance in convexity and order theory.

Contribution

It provides a comprehensive analysis of barycentric algebras, connecting algebraic properties with geometric and physical applications, and illustrating their significance in convexity and ordered structures.

Findings

Barycentric algebras formalize convexity in algebraic terms

Examples demonstrate the applicability in geometry and physics

Applications highlight the importance of barycentric structures in various fields

Abstract

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on various examples and applications, reviewing the pertinence of the barycentric algebra structure.