Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Lipschitz extension from finite subsets   We prove that for every $n\in \mathbb{N}$ there exists a metric space
$(X,d_X)$, an $n$-point subset $S\subseteq X$, a Banach space $(Z,\|\cdot\|_Z)$
and a $1$-Lipschitz function $f:S\to Z$ such that the Lipschitz constant of
every function $F:X\to Z$ that extends $f$ is at least a constant multiple of
$\sqrt{\log n}$. This improves a bound of Johnson and Lindenstrauss. We also
obtain the following quantitative counterpart to a classical extension theorem
of Minty. For every $\alpha\in (1/2,1]$ and $n\in \mathbb{N}$ there exists a
metric space $(X,d_X)$, an $n$-point subset $S\subseteq X$ and a function
$f:S\to \ell_2$ that is $\alpha$-H\"older with constant $1$, yet the
$\alpha$-H\"older constant of any $F:X\to \ell_2$ that extends $f$ satisfies $$
\|F\|_{\mathrm{Lip}(\alpha)}\gtrsim (\log
n)^{\frac{2\alpha-1}{4\alpha}}+\left(\frac{\log n}{\log\log
n}\right)^{\alpha^2-\frac12}. $$ We formulate a conjecture whose positive
solution would strengthen Ball's nonlinear Maurey extension theorem, serving as
a far-reaching nonlinear version of a theorem of K\"onig, Retherford and
Tomczak-Jaegermann. We explain how this conjecture would imply as special cases
answers to longstanding open questions of Johnson and Lindenstrauss and Kalton.

27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg