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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)
##### Keywords:
convex set; separation theorem; cone; natural topology
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