Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Poincar\'{e} cone property   A domain $S\subset{\mathbb{R}}^d$ is said to fulfill the Poincar\'{e} cone
property if any point in the boundary of $S$ is the vertex of a (finite) cone
which does not otherwise intersects the closure $\bar{S}$. For more than a
century, this condition has played a relevant role in the theory of partial
differential equations, as a shape assumption aimed to ensure the existence of
a solution for the classical Dirichlet problem on $S$. In a completely
different setting, this paper is devoted to analyze some statistical
applications of the Poincar\'{e} cone property (when defined in a slightly
stronger version). First, we show that this condition can be seen as a sort of
generalized convexity: while it is considerably less restrictive than
convexity, it still retains some ``convex flavour.'' In particular, when
imposed to a probability support $S$, this property allows the estimation of
$S$ from a random sample of points, using the ``hull principle'' much in the
same way as a convex support is estimated using the convex hull of the sample
points. The statistical properties of such hull estimator (consistency,
convergence rates, boundary estimation) are considered in detail. Second, it is
shown that the class of sets fulfilling the Poincar\'{e} property is a
$P$-Glivenko-Cantelli class for any absolutely continuous distribution $P$ on
$\mathbb{R}^d$. This has some independent interest in the theory of empirical
processes, since it extends the classical analogous result, established for
convex sets, to a much larger class. Third, an algorithm to approximate the
cone-convex hull of a finite sample of points is proposed and some practical
illustrations are given.

15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph