Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Whitney Extension Theorems for convex functions of the classes $C^1$ and
  $C^{1,\omega}$   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 $\omega$. We provide a
necessary and sufficient condition on $f$, $G$ for the existence of a convex
function $F\in C^{1, \omega}(\mathbb{R}^n)$ 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 $K$.

11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin