Daniel Azagra
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View article: $\mathbf{C^2}$-Lusin approximation of strongly convex bodies
$\mathbf{C^2}$-Lusin approximation of strongly convex bodies Open
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…
View article: Approximate Morse-Sard type results for non-separable Banach spaces
Approximate Morse-Sard type results for non-separable Banach spaces Open
For the Banach spaces E=c0(Γ),ℓp(Γ), where Γ is an arbitrary infinite set and 1<p<∞, we show that for every (non-zero) quotient F of E, every continuous function f:E→F can be uniformly approximated by smooth functions with no critical poin…
View article: $\mathbf{C^2}$-Lusin approximation of strongly convex functions
$\mathbf{C^2}$-Lusin approximation of strongly convex functions Open
We prove that if $u:\mathbb{R}^n\to\mathbb{R}$ is strongly convex, then for every $\varepsilon>0$ there is a strongly convex function $v\in C^2(\mathbb{R}^n)$ such that $|\{u\neq v\}|<\varepsilon$ and $\Vert u-v\Vert_\infty<\varepsilon$.
View article: A geometric approach to second-order differentiability of convex functions
A geometric approach to second-order differentiability of convex functions Open
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 con…
View article: A geometric approach to second-order differentiability of convex functions
A geometric approach to second-order differentiability of convex functions Open
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 con…
View article: Inner and outer smooth approximation of convex hypersurfaces. When is it possible?
Inner and outer smooth approximation of convex hypersurfaces. When is it possible? Open
Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We a…
View article: Inner and outer smooth approximation of convex hypersurfaces. When is it possible?
Inner and outer smooth approximation of convex hypersurfaces. When is it possible? Open
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 hyp…
View article: Kirszbraun’s Theorem via an Explicit Formula
Kirszbraun’s Theorem via an Explicit Formula Open
Let $X,Y$ be two Hilbert spaces, let E be a subset of $X,$ and let $G\colon E \to Y$ be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists $\tilde {G} : X \to Y$ with $\tilde {G}=G$ on E and $ \operatorname {\ma…
View article: Extensions of convex functions with prescribed subdifferentials
Extensions of convex functions with prescribed subdifferentials Open
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 contin…
View article: $C^{1,ω}$ extension formulas for $1$-jets on Hilbert spaces
$C^{1,ω}$ extension formulas for $1$-jets on Hilbert spaces Open
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 $F\in C^{1, ω}(X)$. Here $E$ stands for an arbitrary subset of a Hilbert space $X$ and $ω…
View article: $C^{1,\omega}$ extension formulas for $1$-jets on Hilbert spaces
$C^{1,\omega}$ extension formulas for $1$-jets on Hilbert spaces Open
We provide necessary and sufficient conditions for a $1$-jet $(f,\nG):E\\rightarrow \\mathbb{R} \\times X$ to admit an extension $(F, \\nabla F)$ for\nsome $F\\in C^{1, \\omega}(X)$. Here $E$ stands for an arbitrary subset of a\nHilbert sp…
View article: Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,\omega}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces
Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,\omega}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces Open
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 $x_{H}\in C$, a…
View article: Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,ω}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces
Prescribing tangent hyperplanes to $C^{1,1}$ and $C^{1,ω}$ convex hypersurfaces in Hilbert and superreflexive Banach spaces Open
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 $x_{H}\in C$, a…
View article: Subdifferentiable functions satisfy Lusin properties of class $C^1$ or $C^2$
Subdifferentiable functions satisfy Lusin properties of class $C^1$ or $C^2$ Open
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. Assume that for a measurable set $Ω$ and almost every $x\inΩ$ there exists a vector $ξ_x\in\mathbb{R}^n$ such that $$\liminf_{h\to 0}\frac{f(x+h)-f(x)-\langle ξ_x, h\rangle}{|h|^2}>-\infty.$…
View article: Nonsmooth (and smooth, but not very much so) Morse-Sard theorems
Nonsmooth (and smooth, but not very much so) Morse-Sard theorems Open
We prove that every $C^{n-2}$ 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 Le…