Proposition 4

The points of $A$, a convex subset of an affine space $E$ are strictly on the same side of a hyperplane $H$ if, and only if, $A$ does not meet $H$.

Clearly the condition is necessary. Conversely suppose that it is satisfied and let $g(x) = 0$, be an equation defining H ($g$ is an affine linear mapping of E in $\mathbf{R}$). The set $g(A)$ is convex in $\mathbf{R}$, therefore it is an interval, and $0 \notin g(A)$. Hence $g(x)$ is of fixed sign for all $x \in A$.