\begin{definition}[Definition:Minkowski Functional of Open Convex Set]
Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
We define the '''Minkowski functional''' of $C$, $p_C : X \to \hointr 0 \infty$ by: 
:$\map {p_C} x = \inf \set {t > 0 : t^{-1} x \in C}$
for each $x \in X$.
\end{definition}