Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM₁ andM₂ ofX the points on the equidistant hypersurface which have best approximations both inM₁ andM₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.