Geometric properties of the location of subspaces of the space of probability measures

For pairs of subspaces of the space of probability measures defined in an infinite compact set $X$, we examine various geometric and topological properties such as everywhere density, convexity, boundedness, homotopy density, negligibility, and homeomorphism. Also, we establish conditions under which convex, everywhere dense subspaces of the space of probability measures $P(X)$ are boundary sets.