Proposition 3

— In a topological vector space $E$, let $A$ be a closed convex set with a non-empty interior. Then every support hyperplane of $A$ is closed and every frontier point of $A$ belongs to at least one support hyperplane of $A$.

Every support hyperplane of $A$ is closed, since all the points of $A$ are on the same side of the hyperplane (II, p. 15, prop. 17). Also if $x_0$ is a frontier point of $A$, then $x_0$ does not belong to the open non-empty convex set $\overset{\circ}{A}$; after th. 1 of II, p. 36 there exists a hyperplane $H$ that contains $x_0$ and does not meet $\overset{\circ}{A}$. As $A$ is the closure of $\overset{\circ}{A}$ (II, p. 14, cor. 1 to prop. 16), it follows from prop. 17 of II, p. 15 that $H$ is a support hyperplane of $A$.