
By Victor L. (editor) Klee
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Additional info for Convexity (Proceedings of symposia in pure mathematics, Vol.7)
Example text
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!