Properties of the Minkowski Gauge of a Convex Set

Let $X$ be a real vector space and let $\Omega\subseteq X$ be absorbing and convex. Then the Minkowski gauge $p_\Omega$ is finite-valued and sublinear. Moreover,

1. the strict sublevel set is the algebraic interior:
   $$\{x\in X:p_\Omega(x)<1\}=\operatorname{core}(\Omega);$$
2. the non-strict sublevel set is the linear closure:
   $$\{x\in X:p_\Omega(x)\le 1\}=\operatorname{lin}(\Omega).$$