We partly extend the localisation technique from convex geometry to multiple constraints setting. For a given 1-Lipschitz map u:Rn?Rm, m?n, we define and prove existence of partition Rn, up set Lebesgue measure zero, into maximal closed sets such that restriction u is an isometry on these sets. consider disintegration, with respect this partition, log-concave measure. for almost every dimension...

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...