Real Convex Sets and Halfspaces

Main result

Let $(b_i)_{i \in I}$ be a family in $\R ^n$ and $(\beta _i)_{i \in I}$ be a family in $\R $. The set

\[ \Set*{x \in \R ^n}{\ip{x,b_i} \leq \beta _i \text{ for all } i \in I} \]

A polyhedral convex set is one which can be expressed as the intersection of a finite family of closed halfspaces.