Optimization of entropy probability distributions with respect to general closed and convex constraint sets

In this paper, we consider a Differential Entropy Maximization Problem when standard techniques are not suitable. We use a purely topological Fan-KKM Theorem to prove the existence of a maximum entropy probability distribution with general closed and convex constraints for an Effective Counting Entropy functional of a special class of continuous random variables with values in [0,a] such that the probability density functions are nonnegative and have finite second moments.