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The distance function from the boundary in a Minkowski space

Authors:	Graziano Crasta Annalisa Malusa

Institution:	Dipartimento di Matematica ``G. Castelnuovo', Univ. di Roma I, P.le A. Moro 2 -- 00185 Roma, Italy ; Dipartimento di Matematica ``G. Castelnuovo', Univ. di Roma I, P.le A. Moro 2 -- 00185 Roma, Italy

Abstract:	Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure (that is, $M\colon \mathbb{R}^n \to 0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain $\Omega\subset\mathbb{R}^n$ of class , let $d_{\Omega}(x) := \inf\{d^M(x,y); y\in\partial\Omega\}$ be the Minkowski distance of a point $x\in\Omega$ from the boundary of $\Omega$ . We prove that a suitable extension of $d_{\Omega}$ to $\mathbb{R}^n$ (which plays the rôle of a signed Minkowski distance to $\partial \Omega$ ) is of class in a tubular neighborhood of $\partial \Omega$ , and that $d_{\Omega}$ is of class outside the cut locus of $\partial\Omega$ (that is, the closure of the set of points of nondifferentiability of $d_{\Omega}$ in $\Omega$ ). In addition, we prove that the cut locus of $\partial \Omega$ has Lebesgue measure zero, and that $\Omega$ can be decomposed, up to this set of vanishing measure, into geodesics starting from $\partial\Omega$ and going into $\Omega$ 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 $x\in \Omega$ outside the cut locus the pair $(p(x), d_{\Omega}(x))$ , where denotes the (unique) projection of on $\partial\Omega$ , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

Keywords:	Distance function Minkowski structure cut locus Hamilton-Jacobi equations

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