Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Comparison results for capacity   We obtain in this paper bounds for the capacity of a compact set $K$. If $K$
is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth
boundary, and the principal curvatures of $\partial K$ are larger than or equal
to $H_0>0$, then ${\rm Cap}(K)\geq (n-1)\,H_0{\rm vol}(\partial K)$. When $K$
is contained in an $(n+1)$-dimensional manifold with non-negative Ricci
curvature, has smooth boundary, and the mean curvature of $\partial K$ is
smaller than or equal to $H_0$, we prove the inequality ${\rm Cap}(K)\leq
(n-1)\,H_0{\rm vol}(\partial K)$. In both cases we are able to characterize the
equality case. Finally, if $K$ is a convex set in Euclidean space
$\mathbb{R}^{n+1}$ which admits a supporting sphere of radius $H_0^{-1}$ at any
boundary point, then we prove ${\rm Cap}(K)\geq
(n-1)\,H_0\mathcal{H}^n(\partial K)$ and that equality holds for the round
sphere of radius $H_0^{-1}$.

33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics