On the structure of optimal solutions in a mathematical programming problem in a convex space

We consider an optimization problem in a convex space E with an affine objective function, subject to J affine constraints, where J is a given nonnegative integer. We apply the Feinberg-Shwartz lemma in finite dimensional convex analysis to show that there exists an optimal solution, which is in the form of a convex combination of no more than J+1 extreme points of E. The concerned problem does not seem to fit into the framework of standard convex optimization problems