A cone-theoretic barycenter existence theorem

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 $\mathfrak{C}$.