International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1978 / Article

Open Access

Volume 1 |Article ID 629217 | 19 pages | https://doi.org/10.1155/S0161171278000460

Tangent cones, starshape and convexity

Received28 Mar 1978

Abstract

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.

Copyright © 1978 Hindawi Publishing Corporation. 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.


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