Lemma 3

Let E be a Hausdorff locally convex space over $\mathbf{R}$, and K a metrizable compact convex subset of E. Then, there exists a strictly convex numerical function on K.

For, the Banach space $\mathscr{C}(K; \mathbf{R})$ is separable (GT, X, §3, No. 3, Th. 1), therefore so is the subspace $\mathscr{A}$ of $\mathscr{C}(K; \mathbf{R})$ formed by the restrictions to K of the continuous affine linear functions on E. Thus let $(h_n)$ be a sequence dense in $\mathscr{A}$, and let $\alpha_n > \sup_{\mathbf{x} \in K} |h_n(\mathbf{x})|$. Then each of the functions $h_n^2 / n^2 \alpha_n^2$ is convex in K (TVS, II, §2, No. 8, Examples), and the series with general term $h_n^2 / n^2 \alpha_n^2$ is normally convergent, therefore its sum $u$ is continuous and convex in K. It remains to see that $u$ is strictly convex, and for this it suffices to prove that for any two distinct points $\mathbf{x}, \mathbf{x}'$ of K, there is an integer $n$ such that the restriction of $h_n^2$ to the segment with endpoints $\mathbf{x}, \mathbf{x}'$ is strictly convex; but for this it suffices that $h_n(\mathbf{x}) \neq h_n(\mathbf{x}')$ (loc. cit.). Now, there exists a function $h \in \mathscr{A}$ such that $h(\mathbf{x}) \neq h(\mathbf{x}')$ (TVS, II, §4, No. 1, Cor. 1 of Prop. 2) and since the sequence $(h_n)$ is dense in $\mathscr{A}$, there exists an $n$ such that $h_n(\mathbf{x}) \neq h_n(\mathbf{x}')$.

Q.E.D.