A property of nonseparated convex sets

Abstract

Introduction. The theory of convex sets, and more particularly their separation properties, is a very useful tool in the solution of many optimal control problems. In the case of linear problems the theory of convex sets gives an immediate answer, see for instance LaSalle [1] and Neustadt [2]. When the problem is nonlinear it is sometimes useful to study its linearization and to prove that some results obtained for this linearization are indeed valid for the nonlinear problem itself. In many instances this proof can be carried out by elementary topological arguments. Pontryagin et al, pp. 97 and 111, outline such a proof using the concept of topological index. In the present paper we prove a more general result of the same type using Brouwer's fixed point theorem. We shall prove that