Definition 3

— For a subset $A$ of a vector space $E$, a hyperplane $H$ is called a support-hyperplane of $A$, if $H$ contains at least one point of $A$ and all the points of $A$ lie on the same side of $H$.

Let $f$ be a linear form on $E$ that is not identically zero; to say that the hyperplane of the equation $f(x) = \alpha$ is a support hyperplane of $A$ means that $\alpha$ is either the smallest or the largest member of the set $f(A) \subset \mathbf{R}$. In other words, there exists a support hyperplane of $A$ parallel to the hyperplane of equation $f(x) = 0$, if, and only if, one of the bounds of the set $f(A)$ is finite and belongs to $f(A)$.