Michaelʹs problem and weakly infinite-dimensional spaces

Let X be a compact Hausdorff space. Suppose that any multivalued map F : X → Y , where Y is a G δ subset of a Banach space, such that the values of F are convex and closed in Y, has a continuous single-valued selection. Then we prove that X is weakly infinite-dimensional. This provides a partial solution of G δ -problem, posed by Ernest Michael.