Zobrazeno 1 - 10
of 117
pro vyhledávání: '"Azagra, Daniel"'
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 c
Externí odkaz:
http://arxiv.org/abs/2407.05745
Autor:
Azagra, Daniel, Mudarra, Carlos
Publikováno v:
Comptes Rendus. Mathématique, Vol 358, Iss 5, Pp 551-556 (2020)
Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\rightarrow \mathbb{R}$, $G:K\rightarrow X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ suc
Externí odkaz:
https://doaj.org/article/69ca0bab1c8c4681ad092d414830b537
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 $\Vert u-v\Vert_\infty<\varepsilon$.
Externí odkaz:
http://arxiv.org/abs/2311.03481
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
Externí odkaz:
http://arxiv.org/abs/2303.06265
Autor:
Azagra, Daniel, Stolyarov, Dmitriy
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 $
Externí odkaz:
http://arxiv.org/abs/2204.07498
Autor:
Azagra, Daniel, Hajłasz, Piotr
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)
Externí odkaz:
http://arxiv.org/abs/2011.10279
Autor:
Azagra, Daniel, Mudarra, Carlos
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 $\omega$
Externí odkaz:
http://arxiv.org/abs/1912.13317
Autor:
Azagra, Daniel, Mudarra, Carlos
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 $
Externí odkaz:
http://arxiv.org/abs/1911.04166
Autor:
Azagra, Daniel
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 line
Externí odkaz:
http://arxiv.org/abs/1910.12520
Autor:
Azagra, Daniel
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
Externí odkaz:
http://arxiv.org/abs/1905.02127