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- Title
A Uniform Lower Bound on the Norms of Hyperplane Projections of Spherical Polytopes.
- Authors
Kobos, Tomasz
- Abstract
Let K be a centrally symmetric spherical and simplicial polytope, whose vertices form a (4 n) - 1 -net in the unit sphere in R n . We prove a uniform lower bound on the norms of all hyperplane projections P : X → X , where X is the n-dimensional normed space with the unit ball K. The estimate is given in terms of the determinant function of vertices and faces of K. In particular, if N ≥ n 4 n and K = conv { ± x 1 , ± x 2 , ⋯ , ± x N } , where x 1 , x 2 , ⋯ , x N are independent random points distributed uniformly in the unit sphere, then every hyperplane projection P : X → X satisfies an inequality ‖ P ‖ X ≥ 1 + c n N - (2 n 2 + 4 n + 6) (for some explicit constant c n ), with the probability at least 1 - 3 / N .
- Subjects
SPHERICAL projection; POLYTOPES; UNIT ball (Mathematics); NORMED rings; SPHERES
- Publication
Discrete & Computational Geometry, 2023, Vol 70, Issue 1, p279
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-023-00506-z