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Daniel Azagra

Publications and source records attributed to Daniel Azagra.

At least 19 recordsLinked to original sources
arXiv2024

$\mathbf{C^2}$-Lusin approximation of strongly convex bodies

We prove that, if $W \subset \mathbb{R}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V \supset \partial W$ and $\varepsilon>0$, and $V \supset \partial W$ is open, then there exists a $C^2$ locally strongly convex body $W_{\varepsilon, V}$ such that $\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon$ and $\partial W_{\varepsilon, V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\varepsilon, V}$ is strongly convex as well.

math.CA↗
arXiv2023

A geometric approach to second-order differentiability of convex functions

We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by $C^{1,1}$ convex functions and convex bodies.

math.CA↗
arXiv2022

Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

Let $S$ be a convex hypersurface (the boundary of a closed convex set $V$ with nonempty interior) in $\mathbb{R}^n$. We prove that $S$ contains no lines if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U} \subset U\cap \textrm{int}(V) $. We also show that $S$ contains no rays if and only if for every open set $U\supset S$ there exists a real-analytic convex hypersurface $S_{U}\subset U\setminus V$. Moreover, in both cases, $S_U$ can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of $\mathbb{R}^n$, completely characterizing the class of convex functions that can be approximated in the $C^0$-fine topology by smooth convex functions from above or from below. We also provide similar results for $C^1$-fine approximations

math.MG↗
arXiv2020

Lusin-type properties of convex functions and convex bodies

We prove that if $f:\mathbb{R}^n\to\mathbb{R}$ is convex and $A\subset\mathbb{R}^n$ has finite measure, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}$ such that $\mathcal{L}^n(\{x\in A:\, f(x)\neq g(x)\})<\varepsilon$. As an application we deduce that if $W\subset\mathbb{R}^n$ is a compact convex body then, for every $\varepsilon>0$, there exists a convex body $W_{\varepsilon}$ of class $C^{1,1}$ such that $\mathcal{H}^{n-1}\left(\partial W\setminus \partial W_{\varepsilon}\right)< \varepsilon$. We also show that if $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function and $f$ is not of class $C^{1,1}_{\rm loc}$, then for any $\varepsilon>0$ there is a convex function $g:\mathbb{R}^n\to\mathbb{R}$ of class $C^{1,1}_{\rm loc}$ such that $\mathcal{L}^n(\{x\in \mathbb{R}^n:\, f(x)\neq g(x)\})<\varepsilon$ if and only if $f$ is essentially coercive, meaning that $\lim_{|x|\to\infty}f(x)-\ell(x)=\infty$ for some linear function $\ell$. A consequence of this result is that, if $S$ is the boundary of some convex set with nonempty interior (not necessarily bounded) in $\mathbb{R}^n$ and $S$ does not contain any line, then for every $\varepsilon>0$ there exists a convex hypersurface $S_{\varepsilon}$ of class $C^{1,1}_{\textrm{loc}}$ such that $\mathcal{H}^{n-1}(S\setminus S_{\varepsilon})<\varepsilon$.

math.CA↗
arXiv2019

$C^{1,\omega}$ extension formulas for $1$-jets on Hilbert spaces

We provide necessary and sufficient conditions for a $1$-jet $(f, G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ for some $F\in C^{1, \omega}(X)$. Here $E$ stands for an arbitrary subset of a Hilbert space $X$ and $\omega$ is a modulus of continuity. As a corollary, in the particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension $n$. Furthermore, we construct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear) operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) $F$ is given by an explicit formula; (3) $(F, \nabla F)$ depend continuously on the given data $(f, G)$; (4) if $f$ is bounded (resp. if $G$ is bounded) then so is $F$ (resp. $F$ is Lipschitz). We also provide similar results on superreflexive Banach spaces.

math.FA↗
arXiv2019

Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\to\mathbb{R}$, $G:K\to X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ such that $F=f$ on $K$ and $\nabla F=G$ on $K$ is that the $1$-jet $(f, G)$ satisfies (1) $f(x)\geq f(y)+ \langle G(y), x-y\rangle$ for all $x, y\in K$, and (2) if $x, y\in K$ and $f(x)= f(y)+ \langle G(y), x-y\rangle$ then $G(x)=G(y)$. We also solve a similar problem for $K$ replaced with an arbitrary bounded subset of $X$, and for $C^1(X)$ replaced with the class $C^{1,u}_{b}(X)$ of differentiable functions with uniformly continuous derivatives on bounded subsets of $X$.

math.FA↗
arXiv2019

On the global shape of continuous convex functions on Banach spaces

We make some remarks on the global shape of continuous convex functions defined on a Banach space $Z$. Among other results we prove that if $Z$ is separable then for every continuous convex function $f:Z\to\mathbb{R}$ there exist a unique closed linear subspace $Y_f$ of $Z$ such that, for the quotient space $X_f :=Z/Y_{f}$ and the natural projection $\pi:Z\to X_f$, the function $f$ can be written in the form $$ f(z)=\varphi(\pi(z)) +\ell(z) \textrm{ for all } z\in Z, $$ where $\ell_{f}\in X^{*}$ and $\varphi:X_f\to\mathbb{R}$ is a convex function such that $\lim_{t\to\infty}\varphi(x+tv)=\infty$ for every $x, v\in X_f$ with $v\neq 0$. This kind of result is generally false if $Z$ is nonseparable (even in the Hilbertian case $Z=\ell_{2}(\Gamma)$ with $\Gamma$ an uncountable set).

math.FA↗
arXiv2019

Locally $C^{1,1}$ convex extensions of $1$-jets

Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be given functions. We provide necessary and sufficient conditions for the existence of a convex function $F\in C^{1,1}_{\textrm{loc}}(\mathbb{R}^n)$ such that $F=f$ and $\nabla F=G$ on $E$. We give a useful explicit formula for such an extension $F$, and a variant of our main result for the class $C^{1, \omega}_{\textrm{loc}}$, where $\omega$ is a modulus of continuity. We also present two applications of these results, concerning how to find $C^{1,1}_{\textrm{loc}}$ convex hypersurfaces with prescribed tangent hyperplanes on a given subset of $\mathbb{R}^n$, and some explicit formulas for (not necessarily convex) $C^{1,1}_{\textrm{loc}}$ extensions of $1$-jets.

math.FA↗
arXiv2019

Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,\omega}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces

Let $X$ denote $\mathbb{R}^n$ or, more generally, a Hilbert space. Given an arbitrary subset $C$ of $X$ and a collection $\mathcal{H}$ of affine hyperplanes of $X$ such that every $H\in\mathcal{H}$ passes through some point $x_{H}\in C$, and $C=\{x_H : H\in\mathcal{H}\}$, what conditions are necessary and sufficient for the existence of a $C^{1,1}$ convex hypersurface $S$ in $X$ such that $H$ is tangent to $S$ at $x_H$ for every $H\in\mathcal{H}$? In this paper we give an answer to this question. We also provide solutions to similar problems for convex hypersurfaces of class $C^{1, \omega}$ in Hilbert spaces, and for convex hypersurfaces of class $C^{1, \alpha}$ in superreflexive Banach spaces having equivalent norms with moduli of smoothness of power type $1+\alpha$, $\alpha\in (0, 1].$

math.FA↗
arXiv2018

Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets

Let $E$, $F$ be separable Hilbert spaces, and assume that $E$ is infinite-dimensional. We show that for every continuous mapping $f:E\to F$ and every continuous function $\varepsilon: E\to (0, \infty)$ there exists a $C^{\infty}$ mapping $g:E\to F$ such that $\|f(x)-g(x)\|\leq\varepsilon(x)$ and $Dg(x):E\to F$ is a surjective linear operator for every $x\in E$. We also provide a version of this result where $E$ can be replaced with a Banach space from a large class (including all the classical spaces with smooth norms, such as $c_0$, $\ell_p$ or $L^{p}$, $1<p<\infty$), and $F$ can be taken to be any Banach space such that there exists a bounded linear operator from $E$ onto $F$. In particular, for such $E, F$, every continuous mapping $f:E\to F$ can be uniformly approximated by smooth open mappings. Part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from Banach spaces or Hilbert manifolds.

math.FA↗
arXiv2018

Extensions of convex functions with prescribed subdifferentials

Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb{R}$ be a function, and $G:E \rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension $F: X \rightarrow \mathbb{R} $ of $f$ such that the subdifferential $\partial F$ of $F$ coincides with $G$ on $E.$

math.FA↗
arXiv2018

Kirszbraun's theorem via an explicit formula

Let $X,Y$ be two Hilbert spaces, $E$ a subset of $X$ and $G: E \to Y$ a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists $\widetilde{G} : X \to Y$ with $\widetilde{G}=G$ on $E$ and $\textrm{Lip}(\widetilde{G})=\textrm{Lip}(G).$ In this note we show that in fact the function $$\widetilde{G}:=\nabla_Y(\textrm{conv}(g))( \cdot , 0), \qquad \text{where} $$ $$ g(x,y) = \inf_{z \in E} \lbrace \langle G(z), y \rangle + \tfrac{M}{2} \|(x-z,y)\|^2 \rbrace + \tfrac{M}{2}\|(x,y)\|^2, $$ defines such an extension. We apply this formula to get an extension result for {\em strongly biLipschitz homeomorphisms.} Related to the latter, we also consider extensions of $C^{1,1}$ strongly convex functions.

math.FA↗
arXiv2017

Global geometry and $C^1$ convex extensions of $1$-jets

Let $E$ be an arbitrary subset of $\mathbb{R}^n$ (not necessarily bounded), and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $(f,G)$ to have an extension $(F, \nabla F)$ with $F:\mathbb{R}^n\to\mathbb{R}$ convex and of class $C^{1}$. Besides, if $G$ is bounded we can take $F$ so that $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$. As an application we also solve a similar problem about finding convex hypersurfaces of class $C^1$ with prescribed normals at the points of an arbitrary subset of $\mathbb{R}^n$.

math.DG↗
arXiv2017

Explicit formulas for $C^{1, 1}$ and $C^{1, \omega}_{\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces

Given $X$ a Hilbert space, $\omega$ a modulus of continuity, $E$ an arbitrary subset of $X$, and functions $f:E\to\mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F, \nabla F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1, \omega}(X)$, by means of a simple explicit formula. As a consequence of this result, if $\omega$ is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if $X$ is a superreflexive Banach space, we establish similar results for the classes $C^{1, \alpha}_{\textrm{conv}}(X)$.

math.FA↗
arXiv2017

Some remarks about The Morse-Sard theorem and approximate differentiability

Let $n, m$ be positive integers, $n\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function $f:\mathbb{R}^n\to\mathbb{R}^m$ is locally Lipschitz and is approximately differentiable of order $i$ almost everywhere with respect to the Hausdorff measure $\mathcal{H}^{i+m-2}$, for every $i=2, \dots, n-m+1$, then $f$ has the Morse-Sard property (that is to say, the image of the critical set of $f$ is null with respect to the Lebesgue measure in $\mathbb{R}^m$).

math.FA↗
arXiv2016

Nonsmooth Morse-Sard theorems

We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential $\partial_{P}$, we see that for every lower semicontinuous function $f:\mathbb{R}^2\to\mathbb{R}$ the set $f(\{x\in\mathbb{R}^2 : 0\in\partial_{P}f(x)\})$ is $\mathcal{L}^{1}$-null.

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↗