Properties of the Minkowski Gauge of a Convex Set

For absorbing convex Ω, pΩ is sublinear and its level sets describe core(Ω) and lin(Ω).

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:

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