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 $\mathfrak{ C}$.
Categories: math.GN