Gauge (Minkowski functional) of Convex Set

Theorem: Let $K$ be a convex set.

(i) $p_K(x)\leq 1$ if $x\in K$ .

(ii) $p_K(x)<1$ if and only if $x$ is an interior point of $K$ .

We need the following definition: $p_K(x)=\inf\{a\mid a>0,x/a\in K\}$ . This $p_K$ is often known as a Gauge or Minkowski functional.

Proof:

(i) If $x\in K$ , then $x/1\in K$ , so $P_K(x)\leq 1$ . This is the easy part. The converse holds here, if $x\notin K$ , then $p_K(x)>1$ .

(ii) We first prove the “only if” part. Assume $p_K(x)<1$ . Suppose $x$ is not an interior point of $K$ , i.e. there exists $y\in X$ such that for all $\epsilon>0$ , $x+ty\notin K$ for some $|t|<\epsilon$ .

We then have $p_K(x+ty)>1$ . Combining with the subadditive property of the gauge, we have $1<p_K(x+ty)\leq p_K(x)+p_K(ty)$ . Rearranging, we get $p_K(x)>1-p_K(ty)$ . By considering the various possibilities of the sign of $t$ , and using the positive homogeneity of the gauge, we can obtain a contradiction. For example, if $t>0$ , $p_K(x)>1-tp_K(y)$ . Since $t\to 0$ as $\epsilon\to 0$ , this implies $p_K(x)\geq 1$ , a contradiction.