Leaves decompositions in Euclidean spaces

We partly extend the localisation technique from convex geometry to the multiple constraints setting.
For a given 1-Lipschitz map u: R
n → R
m, m ≤ n, we define and prove the existence of a partition of R
n
, up to a
set of Lebesgue measure zero, into maximal closed convex sets such that restriction of u is an isometry on these sets.
We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost
every set of the partition of dimension m, the associated conditional measure is log-concave. This result is proven
also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms
a conjecture of Klartag.