Fixed-point sets of semifree group actions on infinite-dimensional compact convex sets and their manifolds

Let $A$ be the fixed point set of a semifree action of a compact (metric) group $G$ on a Keller space $X$ (an infinite-dimensional compact convex subset of a separable Fr\' echet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all $X$'s semifreely with unique