arXiv subjects

Robert Deville

Publications and source records attributed to Robert Deville.

7 recordsLinked to original sources

arXiv2024

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L f(\bar x)$ of limits of derivatives $\{Df(x_n)\}_{n\geq 1}$ of sequences $\{x_n\}_{n\geq 1}$ converging to~$\bar x$. The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.

math.CA↗

arXiv2020

Normal and starlike tilings in separable Banach spaces

In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a ball of radius r and is contained in a ball of radius R.

math.FA↗

arXiv2020

Normal tilings of a Banach space and its ball

We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give precise constructions of convex normal tilings of the unit sphere, the unit ball or in general of any convex body.

math.FA↗

arXiv2018

Approximation of Lipschitz functions preserving boundary values

Given an open subset $\Omega$ of a Banach space and a Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $\Omega$ by $C^k$-smooth Lipschitz functions which coincide with $u_0$ on the boundary $\partial \Omega$ of $\Omega$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$-Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ defined on the closure $\overline{\Omega}$ of an open subset $\Omega$ of a finite dimensional normed space of dimension $n \geq 2$, and such that the Lipschitz constant of the restriction of $u_0$ to the boundary of $\Omega$ is less than $1$, can be uniformly approximated by differentiable $1$-Lipschitz functions $w$ which coincide with $u_0$ on $\partial \Omega$ and satisfy the equation $\| D w\|_* =1$ almost everywhere on $\Omega.$ This result does not hold in general without assumption on the restriction of $u_0$ to the boundary of $\Omega$.

math.FA↗

arXiv2018

Metric and geometric relaxations of self-contracted curves

Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any $\lambda\in\lbrack-1,1)$, to the metric notion of $\lambda $-curve and the (weaker) geometric notion of $\lambda$-cone property ($\lambda$-eel). In the Euclidean space $\mathbb{R}^{d}$ it is established that for $\lambda\in\lbrack-1,1/d)$ bounded $\lambda$-curves have finite length. For $\lambda\geq 1/\sqrt{5}$ it is always possible to construct bounded curves of infinite length in ${\mathbb{R}}^{3}$ which do satisfy the $\lambda $-cone property. This can never happen in ${\mathbb{R}}^{2}$ though: it is shown that all bounded planar curves with the $\lambda$-cone property have finite length.

math.MG↗

arXiv2016

Lipschitz Embeddings of Metric Spaces into $c_0$

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$\lambda$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<\lambda$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$\lambda$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $\pi(\lambda)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs.

math.FA↗

arXiv2013

A characterization of the Radon-Nikodym property

It is well known that every bounded below and non increasing sequence in the real line converges. We give a version of this result valid in Banach spaces with the Radon-Nikodym property, thus extending a former result of A. Proch\'azka.

math.FA↗