Abstract
AbstractConsider a random simplex $$[X_1,\ldots ,X_n]$$
[
X
1
,
…
,
X
n
]
defined as the convex hull of independent identically distributed (i.i.d.) random points $$X_1,\ldots ,X_n$$
X
1
,
…
,
X
n
in $$\mathbb {R}^{n-1}$$
R
n
-
1
with the following beta density: "Equation missing" Let $$J_{n,k}(\beta )$$
J
n
,
k
(
β
)
be the expected internal angle of the simplex $$[X_1,\ldots ,X_n]$$
[
X
1
,
…
,
X
n
]
at its face $$[X_1,\ldots ,X_k]$$
[
X
1
,
…
,
X
k
]
. Define $${\tilde{J}}_{n,k}(\beta )$$
J
~
n
,
k
(
β
)
analogously for i.i.d. random points distributed according to the beta$$'$$
′
density $${\tilde{f}}_{n-1,\beta } (x) \propto (1+\Vert x\Vert ^2)^{-\beta }, x\in \mathbb {R}^{n-1}, \beta > ({n-1})/{2}.$$
f
~
n
-
1
,
β
(
x
)
∝
(
1
+
‖
x
‖
2
)
-
β
,
x
∈
R
n
-
1
,
β
>
(
n
-
1
)
/
2
.
We derive formulae for $$J_{n,k}(\beta )$$
J
n
,
k
(
β
)
and $${\tilde{J}}_{n,k}(\beta )$$
J
~
n
,
k
(
β
)
which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $$\beta $$
β
. For $$J_{n,1}(\pm 1/2)$$
J
n
,
1
(
±
1
/
2
)
we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope $$K_{n,d} := [U_1,\ldots ,U_n]$$
K
n
,
d
:
=
[
U
1
,
…
,
U
n
]
where $$U_1,\ldots ,U_n$$
U
1
,
…
,
U
n
are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of $$K_{n,d}$$
K
n
,
d
: $$ \lim _{n\rightarrow \infty } n^{-{({d-1})/({d+1})}}{\mathbb {E}}{\mathbf {f}}(K_{n,d}) = {\mathbf {c}}_d \cdot \Omega (K),$$
lim
n
→
∞
n
-
(
d
-
1
)
/
(
d
+
1
)
E
f
(
K
n
,
d
)
=
c
d
·
Ω
(
K
)
,
where $$\Omega (K)$$
Ω
(
K
)
is the affine surface area of K, and $${\mathbf {c}}_d$$
c
d
is an unknown vector not depending on K. We compute $${\mathbf {c}}_d$$
c
d
explicitly in dimensions up to $$d=10$$
d
=
10
and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
Cited by
8 articles.
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