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Angle Sums of Schläfli Orthoschemes.
- Source :
-
Discrete & Computational Geometry . Jul2022, Vol. 68 Issue 1, p125-164. 40p. - Publication Year :
- 2022
-
Abstract
- We consider the simplices 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 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 and K n B . This setting contains sums of external and internal angles of K n A and 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *RANDOM numbers
*CONES
*RANDOM walks
*ANGLES
Subjects
Details
- Language :
- English
- ISSN :
- 01795376
- Volume :
- 68
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Discrete & Computational Geometry
- Publication Type :
- Academic Journal
- Accession number :
- 157099368
- Full Text :
- https://doi.org/10.1007/s00454-021-00326-z