Proposition 22

Let A be a convex set, containing 0, in the vector space E. For all $x \in E$, put

(0 $\leq p_A(x) \leq \infty$). The function $p_A$ satisfies for all $x, y$ in E and $\lambda > 0$. If $V(p_A, \alpha)$ (resp. $W(p_A, \alpha)$) denotes the set of $x \in E$ for which $p_A(x) \leq \alpha$ (resp. $p_A(x) < \alpha$), then If A is absorbent then $p_A$ is finite (therefore sublinear).

Since the relations $x \in \rho A$ and $\lambda x \in \lambda \rho A$ are equivalent when $\lambda > 0$, we have $p_A(\lambda x) = \lambda p_A(x)$ for $\lambda > 0$. Let $x, y$ be two points of E. If $x$ (resp. $y$) is not absorbed by A then $p_A(x) = +\infty$ (resp. $p_A(y) = +\infty$) and the inequality $p_A(x + y) \leq p_A(x) + p_A(y)$ is obviously true. Suppose there exist $\alpha > 0, \beta > 0$ such that $x \in \alpha A$, and $y \in \beta A$; then $x + y \in \alpha A + \beta A = (\alpha + \beta) A$ (II, p. 8, Remark); and thus $p_A(x + y) \leq p_A(x) + p_A(y)$. The inclusion $A \subset V(p_A, 1)$ is clearly true. The inclusion $W(p_A, 1) \subset A$ follows because A is convex and contains 0. Finally if A is absorbent then $p_A$ is obviously finite.

The function $p_A$ defined by (7) is called the gauge of the convex set A. If A is absorbent and symmetric, then $p_A$ is a semi-norm.