习题 22, § 7

By an extremal ray of a convex subset A of a vector space E we mean a closed half line D contained in A, such that, for all $x \in D$ and every open segment with end points $a, b$ in A, and which contains x, it is necessarily the case that $a \in D$ and $b \in D$; the end point of D is an extremal point of A. a) In a Hausdorff locally convex space E, show that every locally compact closed convex set, not containing a line, is the closed convex envelope of the union of its extremal rays and its extremal points. (Suppose the contrary, and writing B for this closed convex envelope, note first that by exerc. 21, d), there exists a closed hyperplane H so that $H \cap A$ is compact and non-empty and $H \cap B = \varnothing$. 然后证明，如果 $a \in H \cap A$ 是 $H \cap A$ 的极值点（因此不是 A 假设的极值点），并且如果结尾为点 $b, c$ 的开线段 S 包含在 A 中且不包含在 H 中，则包含 $a$，则线 D 包含于 S 必然是包含一个线段包含 $a$，且其端点点是 A 的极值点，或者包含包含 $a$ 的极值射线。 b) 证明如果 E 是有限维的，则 E 中每个不包含包含直线的闭凸集合，都是其极值点及其极值射线的并的凸包络（由 E 的维数上的归纳论证）。