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# The Distance Function from the Boundary in a Minkowski Space

Graziano Crasta and Annalisa Malusa
Transactions of the American Mathematical Society
Vol. 359, No. 12 (Dec., 2007), pp. 5725-5759
Stable URL: http://www.jstor.org/stable/20161843
Page Count: 35
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## Abstract

Let the space ℝⁿ be endowed with a Minkowski structure M (that is, M: ℝⁿ → [0, + ∞]) is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class C²), and let $d^{M}(x,y)$ be the (asymmetric) distance associated to M. Given an open domain $\Omega \subset \ ℝ^{n}$ of class C², let $d_{\Omega}(x)\coloneq \text{inf}\{d^{M}(x,y);y\in \partial \Omega \}$ be the Minkowski distance of a point x ∈ Ω from the boundary of Ω. We prove that a suitable extension of $d_{\Omega}$ to ℝⁿ (which plays the rôle of a signed Minkowski distance to ∂Ω) is of class C² in a tubular neighborhood of ∂Ω, and that $d_{\Omega}$ is of class C²[superscript two] outside the cut locus of ∂Ω (that is, the closure of the set of points of nondifferentiability of $d_{\Omega}$ in Ω). In addition, we prove that the cut locus of ∂Ω has Lebesgue measure zero, and that Ω can be decomposed, up to this set of vanishing measure, into geodesics starting from ∂Ω and going into Ω along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point x ∈ Ω outside the cut locus the pair (p(x), $d_{\Omega}(x)$ ), where p(x) denotes the (unique) projection of x on ∂Ω, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

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