Proposition 2

— Let $A$ be a non-empty compact subset of a topological vector space $E$. For every closed hyperplane $H$ in $E$, there exists a support hyperplane of $A$ parallel to $H$.

For, if $f(x) = \gamma$ is an equation of $H$, where $f$ is a continuous linear form in $E$, the restriction of $f$ to $A$ is continuous, therefore bounded and attains its bounds in $A$ (GT, IV, § 6.1, th. 1).

This demonstrates that there exist one or two support hyperplanes of $A$ parallel to $H$; the first case can only arise when $A$ is completely contained in a hyperplane parallel to $H$.