The Strong Convexity Supporting Condition and the Lipschitz Condition for the Metric Projection

We prove that the strong convexity supporting condition is typical for an arbitrary convex compact set in $\mathbb R^n$. It is shown that, in a certain sense for almost all points, the metric projection onto a convex compact set satisfies the Lipschitz condition with Lipschitz constant strictly less than 1. This condition characterizes the strong convexity supporting condition. The linear convergence of the alternating projection method for a convex compact set with the strong convexity supporting condition and for a proximally smooth set is proved under a certain relation between the constant in the strong convexity supporting condition and the proximal smoothness constant.