Minkowski Function (Gauge)

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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.