There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in th...

in this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $x$,  where $x$ is a $c$-distinguished topological space. then, we show that their dual spaces can be identified in a natural way with certain spaces of radon measures.

Given (Mt)t∈R+ and (M ∗ t )t∈R+ respectively a forward and a backward martingale with jumps and continuous parts, we prove that E[φ(Mt + M ∗ t )] is nonincreasing in t when φ is a convex function, provided the local characteristics of (Mt)t∈R+ and (M ∗ t )t∈R+ satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation bounds for random variables admitting a ...

Divergence functions play a central role in information geometry. Given a manifold M, a divergence function D is a smooth, nonnegative function on the product manifold M ×M that achieves its global minimum of zero (with semi-positive definite Hessian) at those points that form its diagonal submanifold ∆M ⊂ M ×M. In this Chapter, we review how such divergence functions induce i) a statistical st...

A C∞ function f on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (f ◦σ)′′ ≥ 0 for every geodesic inM . We shall apply this notion in a variety of ways to the study of manifolds of negative or nonpositive curvature. Convexity has, of course, long been associated with negative curvature, but convex function seem...