  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 $\mathfrak{C}$.
