On a convex set with nondifferentiable metric projection

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 C 1,1 boundary which possesses the same property.Keywords Metric projection · Directional derivative A convex set with smooth boundaryDefine a strictly decreasing sequence of real numbers {α n } n∈N ⊂ (0, π/2] with α 0 = π/2, lim n→∞ α n = 0 and α n+1 < α n + α n+2 2 for all n ∈ N.(1.1) Now we identify R 2 equipped with the Euclidean norm · with C and let A n = e iα n . A beautiful and surprisingly simple example of a nonempty closed convex set for which the directional derivative of the metric projection mapping fails to exist was constructed by Shapiro in [13]. This set is essentially the convex hull J of the collection of points

Navigating Around Convex Sets

“…Equation (3) implies that in the orientation of the parametrization sketched in Figure 1, r(t) is positive. Thus irz ′′ is parallel to z ′ , but actually points in the opposite direction.…”

“…We now give that construction, and outline the reason it works. Our treatment is loosely based on [3], which contains the full details.…”

“…In the light of this, it becomes nontrivial to find a counterexample to the existence of the onesided derivative of the projection. A beautiful and surprisingly simple example of a nonempty closed continuous convex set for which the directional derivative of the metric projection mapping fails to exist was constructed by Shapiro in [16] and finetuned in [3] to become C 1,1 . For some λ ∈ (0, 1), define a sequence of real numbers {α n } n∈N ⊂ (0, π] by…”

“…In Section 3, we will see that this derivative exists for any piecewise linear polygon. Then we will outline the construction from [3] of a convex body in R 2 whose boundary is C 1,1 (C 1 with a Lipschitz derivative) but for which Π ′ + does not exist (Theorem 1). In Sections 4 and 5, the differential equations are constructed that allow us to continually monitor the distance r(t) to a convex body as we navigate around it.…”

“…But we do wish to point out a few directions in which considerable further research has been done. Our own interest (beside [3]) ultimately derives from a closely related problem, namely: Given a Riemannian manifold and two disjoint compact subsets A and B, what are the topological [4,18] and geometric [11] characteristics of the set of points whose distance to A equals their distance to B? Beside their intrinsic interest, these sets have many applications among others in the study of Brillouin zones in quantum mechanics [19].…”