Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality

Abstract

Contents 11 4. Equivalence of the two constraints 14 5. Convergence of the non-contact set 15 6. Semiconcavity 16 7. Convergence of the gradients 21 7.1. Lower semicontinuity 21 7.2. Proof of Theorem 1.3 (T6) 22 Appendix A. Appendix about semiconcavity 22 Appendix B. A counter-example to the equivalence of (1.3) and (1.14) for small m 23 References 25

Numerical computation of the cut locus via a variational approximation of the distance function

“…as an approximation of Cut b (S). This is justified by some theoretical results regarding problem (1.1) obtained in [10], which will be summarized in section 4. Now the set E m,λ can be well approximated using finite elements on a triangulation of the surface S.…”

“…According to [10,Lemma 3.3], we have u m = d b in a neighborhood of b. Therefore, for ε > 0 small enough, we have u m = d b on B(b, 2ε).…”

“…Moreover, as u m = d b on B(b, 2ε), the gradient of u m is O(ε −1 )-Lipschitz on B(b, 2ε)\B(b, ε). According to lemma [10,Proposition 3.4], u m is also locally C 1,1 on S \ {b}. Therefore its gradient is O(ε −1 )-Lipschitz on S \ B(b, ε).…”

“…In section 3, following the strategy of the λ-medial axis, we define a "λ-cut locus" Cut b (S) λ and show that it can be used as an approximation of the complete cut locus. In section 4, we recall the result from [10] which states that the set E m,λ defined above is a good approximation of Cut b (S) λ if m is big enough. In section 5, we discretize problem (1.1) using finite elements, to find a discrete minimizer v h , where h > 0 is the step of the dicretization.…”

Computing the cut locus of a Riemannian manifoldviaoptimal transport

“…A similar strategy was adopted in [34]. To overcome these limitations, in [26] the authors propose a characterization of the cut locus as the limit in the Hausdorff sense of a variationally-defined thawed region around the cut locus. This allows the construction of a convergent finite-element-based numerical approximation of the cut locus which is described and analyzed in [27].…”