The distance function from the boundary in a Minkowski space

Abstract

Let the space R n \mathbb R^n be endowed with a Minkowski structure M M (that is, M : R n → [ 0 , + ∞ ) 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 C 2 C^2 ), and let d M ( x , y ) d^M(x,y) be the (asymmetric) distance associated to M M . Given an open domain Ω ⊂ R n \Omega \subset \mathbb R^n of class C 2 C^2 , let d Ω ( x ) := inf { d M ( x , y ) ;   y ∈ ∂ Ω } d_{\Omega }(x) := \inf \{d^M(x,y);\ y\in \partial \Omega \} be the Minkowski distance of a point x ∈ Ω x\in \Omega from the boundary of Ω \Omega . We prove that a suitable extension of d Ω d_{\Omega } to R n \mathbb R^n (which plays the rôle of a signed Minkowski distance to ∂ Ω \partial \Omega ) is of class C 2 C^2 in a tubular neighborhood of ∂ Ω \partial \Omega , and that d Ω d_{\Omega } is of class C 2 C^2 outside the cut locus of ∂ Ω \partial \Omega (that is, the closure of the set of points of nondifferentiability of d Ω d_{\Omega } in Ω \Omega ). In addition, we prove that the cut locus of ∂ Ω \partial \Omega has Lebesgue measure zero, and that Ω \Omega can be decomposed, up to this set of vanishing measure, into geodesics starting from ∂ Ω \partial \Omega and going into Ω \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 ∈ Ω x\in \Omega outside the cut locus the pair ( p ( x ) , d Ω ( x ) ) (p(x), d_{\Omega }(x)) , where p ( x ) p(x) denotes the (unique) projection of x x on ∂ Ω \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.