Minimal regularity ensuring differentiability of the return map

Determine the minimal regularity assumptions on the thickness function d: ∂C → R+ and on the convex core boundary ∂C under which the return map F = π ∘ Φ: ∂C → ∂C, with Φ(c) = c + d(c) ν(c) and π the reciprocal map along inward normals to ∂Ω = Φ(∂C), is of class C1. In particular, ascertain whether d ∈ C1,1(∂C) with ∂C of class C2 suffices to ensure that the outer boundary ∂Ω is locally C1,1, that the reciprocal map π is well-defined with the needed regularity, and consequently that F is C1.

Background

The paper shows that if ∂C is C2 and the thickness function d is C2 on ∂C, then the boundary ∂Ω parametrized by Φ(c) = c + d(c)ν(c) has sufficient regularity to make the return map F = π ∘ Φ differentiable (C1). This uses prior regularity results for admissible domains and the reciprocal map.

However, the authors note that these assumptions may be stronger than necessary. They pose the problem of identifying the minimal regularity of d and the associated consequences for ∂Ω and π that would still guarantee F ∈ C1. They specifically suggest the plausibility that d ∈ C1,1(∂C) might suffice, but emphasize that a proof would require a careful analysis of the reciprocal map’s regularity at this lower level of smoothness.

References

A refined analysis of minimal regularity conditions for the differentiability of F remains an interesting open problem. For instance, one may conjecture that d∈ C{1,1}(∂C) (i.e., Lipschitz gradient) is enough to guarantee that ∂Ω is C{1,1} and that F is C{1}, but a rigorous proof would require a careful study of the regularity of the reciprocal map under such minimal hypotheses.

— Geometry, Dynamics and Topology of Thickness Landscape: A Morse-Theoretic Analysis of the Return-Map in the Class $\mathcal{O}_{C}$  (2603.30010 - Barkatou et al., 31 Mar 2026) in Remark (On optimal regularity), Section 7.1 (Differentiability of the return map)