International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 459-477
Tangent cones, starshape and convexity
Department of Mathematics, Dalhousie University, Nova Scotia, Halifax B3H 4H8, Canada
Received 28 March 1978
Copyright © 1978 J. M. Borwein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.