A cone-theoretic barycenter existence theorem

TL;DR

This paper proves that every continuous valuation on certain topological cones has a unique barycenter, and the barycenter map forms a continuous algebraic structure, extending valuation theory in topological cones.

Contribution

It establishes the existence and uniqueness of barycenters for continuous valuations on locally convex, sober topological cones, and characterizes the barycenter map as a continuous algebraic structure.

Findings

Every continuous valuation on the cone has a unique barycenter.

The barycenter map is continuous and forms a $ extbf{V}_w$-algebra.

The barycenter map induces the cone structure uniquely.

Abstract

We show that every continuous valuation on a locally convex, locally convex-compact, sober topological cone $\mathfrak{C}$ has a barycenter. This barycenter is unique, and the barycenter map $\beta$ is continuous, hence is the structure map of a $\mathbf V_{\mathrm w}$-algebra, i.e., an Eilenberg-Moore algebra of the extended valuation monad on the category of $T_0$ topological spaces; it is, in fact, the unique $\mathbf V_{\mathrm w}$-algebra that induces the cone structure on $\mathfrak{C}$.