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 $J$ constraints in the forms of inequalities on some other affine functions, where $J$ is a given nonnegative integer. Under suitable conditions, 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 mixture of no more than $J+1$ extreme points of $E$. It seems that in the current setup, this result has not yet been made available, because the concerned problem does not fit into the framework of standard convex optimization problems.