Exercise 28, § 7

Let $I$ be a non enumerable infinite set. Show that every cap of the cone $\mathbf{R}_+^I$ in $\mathbf{R}^I$ is contained in the sum of subspaces of the form $\mathbf{R}^J$, where $J$ is an enumerable subset of $I$ (use prop. 4 of II, p. 38). Deduce that there are points of $\mathbf{R}_+^I$ which do not belong to any cap of $\mathbf{R}_+^I$, even though $\mathbf{R}_+^I$ is the convex closed envelope of the union of its extremal generators.