Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets

We study the dependence of metric projections on the following three parameters: the point projected, the set to which we are projecting, and the norm (generally speaking, non-symmetric) that determines the metric. We obtain sharp estimates for the moduli of continuity of metric projections onto convex and weakly convex sets in Banach spaces. We also estimate these moduli in terms of the moduli of convexity and smoothness of the space (or the quasi-ball).