Robert Deville

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

AbstractA closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here:(1)LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and let ε>0. Then there exists a tangential ε-approximating polytopeP for the bodyW.(2)LetP be a polytope in a separable Banach spaceE. Then, for every ε>0,P can be ε-approximated by an analytic, closed, convex and bounded bodyV. We deduce from these two results that in a polyhedral Banach space (for instance in c0(ℕ) or inC(K) forK countable compact), every equivalent norm can be approximated by norms which are analytic onE/{0}.

Porosity of ill-posed problems

We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is σ-porous.

Chapter 10 – Perturbed Minimization Principles and Applications

Given a bounded below, lower semi-continuous function f on an infinite dimensional Banach space or a non-compact manifold X , we consider various possibilities of perturbing f by an element p of a reasonable class of functions A in such a way that for the new functional f- p , the minimization problem inf X (f- p ) is well-posed (i.e., every minimizing sequence is convergent).

Stability of subdifferentials of nonconvex functions in Banach spaces

We investigate various notions of subdifferentials and superdifferentials of nonconvex functions in Banach spaces. We prove stability results of these subdifferentials and superdifferentials under various kind of convergences. Our proofs rely on a recent variational principle of Deville, Godefroy and Zizler. Connections between our results, the geometry of Banach spaces and existence theorems of viscosity solutions for first and second-order Hamilton-Jacobi equations in infinite-dimensional Banach spaces will be explained.

The subdifferential of the sum of two functions in Banach spaces. II. Second order case

We prove a formula for the second order subdifferential of the sum of two lower semi continuous functions in finite dimensions. This formula yields an Alexandrov type theorem for continuous functions. We derive from this uniqueness results of viscosity solutions of second order Hamilton-Jacobi equations and singlevaluedness of the associated Hamilton-Jacobi operators. We also provide conterexamples in infinite dimensional Hilbert spaces.

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml1 into ℝ2 which is Gâteaux-differentiable at every point and such that for everyx, y εl1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0.

Stability of Distributed Algorithms in the Face of Incessant Faults

For large distributed systems built from inexpensive components, one expects to see incessant failures. This paper proposes two models for such faults and analyzes two well-known self-stabilizing algorithms under these fault models. For a small number of processes, the properties of interest are verified automatically using probabilistic model-checking tools. For a large number of processes, these properties are characterized using asymptotic bounds from a direct Markov chain analysis and approximated by numerical simulations.

A note on cotype of smooth spaces

For sufficiently smooth Banach spaces weak cotype 2 implies Hilbert space.

Almost classical solutions of Hamilton-Jacobi equations

We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of R(d) or on d-dimensional manifolds whenever d >= 2. In particular, when M is a Riemannian manifold, we prove the existence of a differentiable function a on M which satisfies the Eikonal equation parallel to del u(x)parallel to(x) = 1 almost everywhere on M.

Metric and Geometric Relaxations of Self-Contracted Curves

The metric notion of a self-contracted curve (respectively, self-expanded curve, if we reverse the orientation) is hereby extended in a natural way. Two new classes of curves arise from this extension, both depending on a parameter, a specific value of which corresponds to the class of self-expanded curves. The first class is obtained via a straightforward metric generalization of the metric inequality that defines self-expandedness, while the second one is based on the (weaker) geometric notion of the so-called cone property (eel-curve). In this work, we show that these two classes are different; in particular, curves from these two classes may have different asymptotic behavior. We also study rectifiability of these curves in the Euclidean space, with emphasis in the planar case.