Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Holes and islands in random point sets   For $d\in\mathbb{N}$, let $S$ be a set of points in $\mathbb{R}^d$ in general
position. A set $I$ of $k$ points from $S$ is a $k$-island in $S$ if the convex
hull $\mathrm{conv}(I)$ of $I$ satisfies $\mathrm{conv}(I) \cap S = I$. A
$k$-island in $S$ in convex position is a $k$-hole in $S$.
  For $d,k\in\mathbb{N}$ and a convex body $K\subseteq\mathbb{R}^d$ of volume
$1$, let $S$ be a set of $n$ points chosen uniformly and independently at
random from $K$. We show that the expected number of $k$-holes in $S$ is in
$O(n^d)$. Our estimate improves and generalizes all previous bounds. In
particular, we estimate the expected number of empty simplices in $S$ by
$2^{d-1}\cdot d!\cdot\binom{n}{d}$. This is tight in the plane up to a
lower-order term.
  Our method gives an asymptotically tight upper bound $O(n^d)$ even in the
much more general setting, where we estimate the expected number of $k$-islands
in $S$.

21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol