Exercise 25, § 2

a) Show that, in a topological vector space, every convex set with an interior point, is such that its frontier is nowhere dense (use prop. 16 of II, p. 14). b) In a Hausdorff topological vector space $E$, let $A$ be a closed convex set with an interior point, and let $H$ be a closed hyperplane that contains an interior point of $A$. Show that the intersection of $H$ and of the frontier $F$ of $A$ is a set which is nowhere dense relative to $F$ (to show that in every neighbourhood of a point of $H \cap F$ there exist points of $F$ not in $H$, reduce to the case when $E$ is of dimension 2).