Proposition 2

— Let K be a compact convex subset of a Hausdorff locally convex space E, $\mu$ a positive measure of total mass 1 on K, $b_\mu$ its barycenter. For every convex numerical function $f \geq 0$ that is lower semi-continuous on K,

It is known (TVS, II, §5, No. 4, Prop. 5) that $f$ is the upper envelope of a family of restrictions to K of continuous affine linear functions $h_\alpha : x \mapsto c_\alpha + \langle x, z'_\alpha \rangle$. Then for every $\alpha$; now, $\int h_\alpha(x) d\mu(x) = c_\alpha + \int \langle x, z'_\alpha \rangle d\mu(x)$ since $\mu$ has total mass 1; but $\int \langle x, z'_\alpha \rangle d\mu(x) = \langle b_\mu, z'_\alpha \rangle$ by the definition of barycenter. Therefore $\sup_\alpha \int h_\alpha(x) d\mu(x) = \sup_\alpha h_\alpha(b_\mu) = f(b_\mu)$, whence the conclusion.

When $\mu$ is a discrete positive measure on K of total mass 1, Prop. 2 yields anew the inequality that defines the convex functions on K.