Convexity (Proceedings of symposia in pure mathematics, by Victor L. (editor) Klee

By Victor L. (editor) Klee

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L. Klee, Research problem No. 5, Bull. Amer. Math. Soc. 63 (1957), 419. 2. T. J. McMinn, On the tine segments of a convex surface in E5, Pacific J. Math. 10 (1960), 943-946. UNIVERSiTY OF PENNSYLVANIA OF A CONVEX SET THE SUPPORT BY ERRETT BISHOP AND R. R. PHELPS The following well-known separation theorem is basic to the considerations of this paper. SEPARATION THEOREM - Suppose that A and B are convex subsets of a real Hausdorff topological vector space E, and that the interior of B is nonempty and disjoint from A.

The space E x R is normable, C x [0, 1[ is homeomorphic with C x [—1, oo[ and C x [0, 1) with C x [—1, 1]. The sets C x [—1, co[ and C x [—1, 1) are closed convex bodies in E x R which have (0,0) as an interior point. The characteristic cone of C x [—1, oo[ is not a linear subspace and (since C contains no line) that of C x [—1, 1] is either not a linear subspace or is equal to ((0,0)), a subspace of infinite deficiency. 3. The following remark is easily verified: TOPOLOGICAL CLASSIFICATION OF CONVEX SETS If F1 and F2 are closed linear subspaces of the same finite deficiency in a topological linear sPace E, there exists a linear of E onto E which carries F1 onto F2.

A convex body is a convex set which has an interior point. By the principal result of § 1, the topological classification problem for closed convex bodies in a normed linear space E is reduced to that for E's unit cell {x E: II XII i} and E's closed linear subspaces of finite deficiency. A corollary asserts that if E admits (for each finite n) a closed linear subspace of deficiency n which is homeomorphic with its own unit cell, then E is homeomorphic with all its closed convex bodies. The support of a real-valued function!

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