The Lipschitz continuity of the distance function to the cut locus

By Jin-ichi Itoh, Minoru Tanaka. 2000

119 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5512eff36f8f33bcbde60b911876108fa3702e67

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In Applied Mathematics

In Astronomy and Astrophysics

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Citations in the corpus

A symmetry problem in the calculus of variations (2006, background): “It is known that if∂Ω is of classC2,1, thenτ is Lipschitz continuous on ∂Ω (see [18, 17]); for less regular domains the Lipschitz continuity may fail, but continuity is preserved if ∂Ω is of classC2 (see [5, 6]).”
A Morse-Sard theorem for the distance function on Riemannian manifolds (2004, methodology): “In [7], Itoh and Tanaka proved the following (we refer the reader to [7] for the definition of the distance function to the cut locus ρ(·) whenever N is an arbitrary submanifold of M).”
Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2 (2009, background): “Up to now it was not known whether the convexity of the injectivity domains is stable under small perturbations of the metric, except in the nonfocal case (see [7, 15, 20]).”
The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton‐Jacobi equations (2003, unknown): “It is not difficult to see that u(x) = inf y∈∂Ω L(x, y) = inf y∈∂Ω dist(y to x), x ∈ Ω.”
Nonlinear evolution by mean curvature and isoperimetric inequalities (2006, methodology): “We now want to apply a result of Li and Nirenberg [17] or equivalently Itoh and Tanaka [15], which says: Given a bounded open set S ⊂ Rn+2 with smooth boundary, define G to be the largest open subset of S such that every point x in G has a unique closest point on ∂S.”
Nearly Round Spheres Look Convex (2012, background): “3] (see also [6, 21]), for any x ∈M we have Hn−1[K ∩ cut(x)]<+∞, (6.”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, background): “We notice that, for every x ∈M , the function tC(x, ·) : UxM → R is locally Lipschitz (see [4, 18, 20]) while the function tF (x, ·) : UxM → R is locally semiconcave on its domain (see [4]).”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Furthermore, having only the bound λ ≤ detD2u ≤ 1/λ is not even enough for C1regularity: the function u(x1, x ′) := |x′|+ |x′|n/2(1 + x21), n ≥ 3, is merely Lipschitz and solves λ ≤ detD2u ≤ 1/λ in a small convex neighborhood of the origin.”
A geometric characterization of a sharp Hardy inequality (2011, result): “[26, 27] The map m(z) and the function ρ̄(z) are in C loc (∂Ω).”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, result): “Σ = ⋃ {t} × cut(yt) ⊂ [0, 1] ×M has finite Hn measure in view of the results in [13, 18].”
THE DISTANCE FUNCTION FROM THE BOUNDARY IN A MINKOWSKI SPACE (2006, methodology): “In the same setting, Itoh and Tanaka [14] have proven that, if Ω is of class C, then the function l is Lipschitz continuous on ∂Ω.”