  We study fine differentiability properties of horizons. We show that the set
of end points of generators of a n-dimensional horizon H (which is included in
a (n+1)-dimensional space-time M) has vanishing n-dimensional Hausdorff
measure. This is proved by showing that the set of end points of generators at
which the horizon is differentiable has the same property. For 1\le k\le n+1 we
show (using deep results of Alberti) that the set of points where the convex
hull of the set of generators leaving the horizon has dimension k is ``almost a
C^2 manifold of dimension n+1-k'': it can be covered, up to a set of vanishing
(n+1-k)-dimensional Hausdorff measure, by a countable number of C^2 manifolds.
We use our Lorentzian geometry results to derive information about the fine
differentiability properties of the distance function and the structure of cut
loci in Riemannian geometry.
