Properties of the Minkowski Gauge of a Convex Set

Let $X$ be a vector space and let $\Omega\subset X$ be absorbing and convex. Consider the Minkowski gauge $p_\Omega$.

Theorem:

1. $p_\Omega$ is real-valued and sublinear.
2. The strict sublevel set equals the algebraic interior (core):
   $$\{x\in X\mid p_\Omega(x)<1\}=\operatorname{core}(\Omega).$$
3. The non-strict sublevel set equals the linear closure:
   $$\{x\in X\mid p_\Omega(x)\le 1\}=\operatorname{lin}(\Omega).$$

Context: This theorem connects the algebraic geometry of $\Omega$ (core/lin) with a canonical sublinear functional. It is the key bridge to Hahn–Banach separation results such as separating a point from a convex set.