On a Convex Set with Nondifferentiable Metric Projection

TL;DR

This paper constructs a convex set with a smooth boundary in the Euclidean plane where the metric projection mapping's directional derivative does not exist, challenging assumptions about smoothness and differentiability.

Contribution

It revisits and modifies Shapiro's example to create a convex set with a smooth boundary exhibiting nondifferentiability of the metric projection.

Findings

Existence of a convex set with smooth boundary lacking directional differentiability of the metric projection

Modification of previous constructions to achieve smoothness

Insights into the relationship between boundary smoothness and projection differentiability

Abstract

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property.