Lemma 1

Let E be the $n$-dimensional numerical space and let A be a convex compact subset of E of which 0 is an interior point. For every $x\in E$, let us denote by $p_A(x)$ the greatest lower bound in $\overline{\mathbf{R}}$ of the set of real numbers $t >0$ such that $x\in tA$.

The mapping $p_A$ is finite, continuous, and has the following properties:

(i) For every $x\in E$ such that $x= 0\not$ , one has $p_A(x)>0$ ;

(ii) For every $s\in \mathbf{R}_+$ and every $x\in E$, one has $p_A(sx) =sp_A(x)$.

(iii) For all $x$ and $y\in E$, one has $p_A(x+y)\leqslant p_A(x) +p_A(y)$.

(iv) In order that a point $x$ of E belong to A, it is necessary and sufficient that $p_A(x)\leqslant 1$.

For $x\in E$, let us denote by $\|x\|$ the euclidean norm of $x$ (TG, VI, p. 7). Since A is compact, there exists a real number $M>0$ such that every point $x$ of A satisfies $\|x\|\leqslant M$. Since 0 is an interior point of A, there exists a real number $m >0$ such that every point $x$ of E such that $\|x\|\leqslant m$ belong to A. Consequently, one has the relation $\|x\|/M\leqslant p_A(x)\leqslant \|x\|/m$, for every $x\in E$. In particular, $p_A(0) = 0$ and $p_A(x)= 0\not$ if $x= 0\not$ .

Assertions (ii) and (iv) follow immediately from the definition of the mapping $p_A$.

Let $x$ and $y$ be points of E. Let $x'$ and $y'$ be points of A such that $x=p_A(x)x'$ and $y=p_A(y)y'$. If $x$ and $y$ are not both zero, $p_A(x) +p_A(y)>0$ and one has

Since A is convex, this proves that $x+y$ belongs to $(p_A(x) +$ $p_A(y))A$, whence $p_A(x+y)\leqslant p_A(x) +p_A(y)$. If $x=y= 0$, this inequality still holds, for $p_A(0) = 0$. This proves assertion (iii).

Applying this inequality to $x+y$ and $-y$, one deduces that, for every pair $(x, y)$ of points of E, one has

$|p_A(x+y)-p_A(x)|\leqslant$ max($p_A(y), p_A(-y)$)$\leqslant m^{-1}\|y\|$.

This proves that the mapping $p_A$ is continuous, whence the lemma.

The mapping $p_A$ is called the gauge of the convex subset A.