Minkowski Function (Gauge)

A set-generated sublinear functional pΩ(x)=inf{t≥0 : x∈tΩ}.

Minkowski Function (Gauge)

Let $X$ be a vector space and let $\Omega\subset X$ be nonempty.

The Minkowski function (or Minkowski gauge) of $\Omega$ is the function $p_\Omega:X\to[0,\infty]$ defined by

p_\Omega(x):=\inf\{t\ge 0 \mid x\in t\Omega\}, \qquad x\in X,

with the convention $\inf(\emptyset)=\infty$ .

When $\Omega$ is absorbing and convex , the gauge is real-valued and sublinear ; its strict and non-strict sublevel sets recover core(Ω) and lin(Ω) via the main Minkowski gauge theorem .

Examples:

If $\Omega$ is the closed unit ball of a norm , then $p_\Omega(x)=\|x\|$ .
If $\Omega$ is a cone (e.g., $\mathbb{R}^n_+$ ), then $p_\Omega$ can take the value $\infty$ outside the cone unless $\Omega$ is absorbing.