Proposition 4

— Let C be a pointed convex cone with vertex 0 in E, a Hausdorff locally convex space.

a) Let A be a cap of C. Let p be the restriction to C of the gauge of A (II, p. 20). The set of the x ∈ C such that p(x) ≤ 1 is the set A. The function p is lower semi-continuous and has the following properties : (i) For any x, y in C, we have p(x + y) = p(x) + p(y). (ii) For any x ∈ C and λ ∈ R_+^*, we have p(λx) = λp(x). (iii) If x ∈ C, the relation p(x) = 0 is equivalent to x = 0.

b) Conversely, let p be a function defined in C with values in [0, +∞], satisfying the conditions (i), (ii) of a). Let A be the set of the x ∈ C such that p(x) ≤ 1. Then A and C − A are convex. A is a cap, if and only if A is compact and non-empty.

The statement b) is obvious. The properties stated in a) are consequences of the remarks preceding prop. 4 and of the prop. 22 of II, p. 20 and prop. 23 of II, p. 20 with the exception of

It is sufficient to prove this last when x ≠ 0 and y ≠ 0; we have therefore p(x) > 0, p(y) > 0. Let μ, λ be two numbers > 0 such that λ < p(x), μ < p(y), and denote the complement of A in C by B. We have x ∈ λB, y ∈ μB, therefore x + y ∈ λB + μB; by the convexity of B, we have λB + μB ⊂ (λ + μ)B, whence p(x + y) > λ + μ, which implies the inequality stated above.