Convexity of open balls in a metrizable subset of a locally convex t.v.s. The following result is well known (Rudin, Functional Analysis, Theorem 1.24): If $X$ is a locally convex topological vector space with a countable local base then there is a compatible metric $d$ such that all open balls are convex.
My question is: Assume that $X$ is a locally convex t.v.s. which is not metrizable and $C$ is a compact, convex and metrizable subset of $X$. Is there a metric compatible with the topology of $C$ such that all open balls are convex?

$Y=\operatorname{span}(C)$ is a locally convex tvs which is still metrisable I think (using that $C$ has a countable base, being compact metrisable). So the first result would apply there.
