International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 459-477

Tangent cones, starshape and convexity

J. M. Borwein

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 [14] have been made. These began with a paper by Edelstein and Keener [8] and have culminated in the papers by Borwein, Edelstein and O'Brien [3] [4] by Edelstein, Keener and O'Brien [9] and finally by O'Brien [16].

Unrelatedly, Borwein and O'Brien [5] posed a question which arises in optimization [2] [11] 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 [16] with attendant strengthening of the results in [5] [16]. This in turn leads to some interesting characterizations of convexity, starshape and of various functional conditions.