arXiv subjects

Carlos Mudarra

Publications and source records attributed to Carlos Mudarra.

21 records · Page 2Linked to original sources

arXiv2016

Whitney Extension Theorems for convex functions of the classes $C^1$ and $C^{1,ω}$

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $ω$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\in C^{1, ω}(\mathbb{R}^n)$ such that $F=f$ on $C$ and $\nabla F=G$ on $C$, with a good control of the modulus of continuity of $\nabla F$ in terms of that of $G$. On the other hand, assuming that $C$ is compact, we also solve a similar problem for the class of $C^1$ convex functions on $\mathbb{R}^n$, with a good control of the Lipschitz constants of the extensions (namely, $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$). Finally, we give a geometrical application concerning interpolation of compact subsets $K$ of $\mathbb{R}^n$ by boundaries of $C^1$ or $C^{1,1}$ convex bodies with prescribed outer normals on $K$.

math.CA↗

arXiv2016

An Extension Theorem for convex functions of class $C^{1,1}$ on Hilbert spaces

Let $\mathbb{H}$ be a Hilbert space, $E \subset \mathbb{H}$ be an arbitrary subset and $f: E \rightarrow \mathbb{R}, \: G: E \rightarrow \mathbb{H}$ be two functions. We give a necessary and sufficient condition on the pair $(f,G)$ for the existence of a \textit{convex} function $F\in C^{1,1}(\mathbb{H})$ such that $F=f$ and $\nabla F =G$ on $E$. We also show that, if this condition is met, $F$ can be taken so that $\textrm{Lip}(\nabla F) = \textrm{Lip}(G)$. We give a geometrical application of this result, concerning interpolation of sets by boundaries of $C^{1,1}$ convex bodies in $\mathbb{H}$. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.

math.FA↗

arXiv2014

Global approximation of convex functions by differentiable convex functions on Banach spaces

We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by $C^k$ smooth convex functions.

math.FA↗