Tangent cones, starshape and convexity
In the last few years various infinite dimensional extensions to Krasnoselski's Theorem on starshaped sets  have been made. These began with a paper by Edelstein and Keener  and have culminated in the papers by Borwein, Edelstein and O'Brien   by Edelstein, Keener and O'Brien  and finally by O'Brien .Unrelatedly, Borwein and O'Brien  posed a question which arises in optimization   of when a closed set is pseudoconvex at all its members.In this paper we show that these two questions can be handled simultaneously through a slight refinement of the powerful central result in  with attendant strengthening of the results in  . This in turn leads to some interesting characterizations of convexity, starshape and of various functional conditions.
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