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On a class of convex sets. (English) Zbl 0618.52001
Let X be a vector space, x a point in X and C a convex subset of X. The author gives necessary and/or sufficient conditions for the equality \((E)\quad X= C + Rx\) to be true. If C is a cone, then (E) holds iff a) C is a linear subspace of codimension one and x is not in C, or b) at least one of x and -x lies in \(C^ i\) \((=\) the core of C). If (E) holds, then exactly one of the following assertions holds: a) there is a linear subspace Y of codimension one such that \(C=c+Y\) and x is not in Y; b) there is a linear subspace Y of codimension one such that, for some real numbers \(a<b\), \(C^ i=\) \((a,b)x+Y\) and x is not in Y; c) \(\bar C+(0,\infty)x\subset C^ i;\) d) \(\bar C+(-\infty,0)x\subset C^ i\) (\(\bar C =\) the closure of C in the natural topology of X).
Reviewer: J.Danes

MSC:
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
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