Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Expected $f$-vector of the Poisson Zero Polytope and Random Convex Hulls
  in the Half-Sphere   We prove an explicit combinatorial formula for the expected number of faces
of the zero polytope of the homogeneous and isotropic Poisson hyperplane
tessellation in $\mathbb R^d$. The expected $f$-vector is expressed through the
coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2
x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and the
expected volume of the spherical convex hull of $n$ random points sampled
uniformly and independently from the $d$-dimensional half-sphere. In the case
when $n=d+2$, we compute the probability that this spherical convex hull is a
spherical simplex, thus solving an analogue of the Sylvester four-point problem
on the half-sphere.

11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph