Proposition 3

In the affine space $E$, let $H$ be a hyperplane defined by the relation $g(x) = 0$, where $g$ is a non-constant affine function on $E$. Then the half-spaces defined by the relations $g(x) \geq 0, g(x) \leq 0, g(x) > 0, g(x) < 0$ are convex.

For these are the inverse images under $g$ of intervals of $\mathbf{R}$ and thus are convex.

With the notations of prop. 3 the points of a subset $M$ of an affine space are on the same side (resp. strictly on the same side) of the hyperplane $H$ if $M$ is contained in one of the half-spaces defined by $g(x) \geq 0, g(x) \leq 0$ (resp. $g(x) > 0$ or $g(x) < 0$).