Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Smooth convex extensions of convex functions   Let $C$ be a compact convex subset of $\mathbb{R}^n$, $f:C\to\mathbb{R}$ be a
convex function, and $m\in\{1, 2, ..., \infty\}$. Assume that, along with $f$,
we are given a family of polynomials satisfying Whitney's extension condition
for $C^m$, and thus that there exists $F\in C^{m}(\mathbb{R}^n)$ such that
$F=f$ on $C$. It is natural to ask for further (necessary and sufficient)
conditions on this family of polynomials which ensure that $F$ can be taken to
be convex as well. We give a satisfactory solution to this problem in the case
$m=\infty$, and also less satisfactory solutions in the case of finite $m\geq
2$ (nonetheless obtaining an almost optimal result for $C$ a finite
intersection of ovaloids). For a solution to a similar problem in the case
$m=1$ (even for $C$ not necessarily convex), see arXiv:1507.03931,
arXiv:1706.09808, arXiv:1706.02235.

8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn