AbstractWe consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$
K
n
A
=
{
x
∈
R
n
+
1
:
x
1
≥
x
2
≥
⋯
≥
x
n
+
1
,
x
1
-
x
n
+
1
≤
1
,
x
1
+
⋯
+
x
n
+
1
=
0
}
and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$
K
n
B
=
{
x
∈
R
n
:
1
≥
x
1
≥
x
2
≥
⋯
≥
x
n
≥
0
}
,
which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$
K
n
A
and $$K_n^B$$
K
n
B
. This setting contains sums of external and internal angles of $$K_n^A$$
K
n
A
and $$K_n^B$$
K
n
B
as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.
Lipschitz Normally Embedded Set and Tangent Cones at Infinity
