Monotonicity of expected \(f\)-vectors for projections of regular polytopes. (English) Zbl 1382.52004
Summary: Let \( P_n\) be an \( n\)-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project \( P_n\) onto a random, uniformly distributed linear subspace of dimension \( d\geq 2\). We prove that the expected number of \( k\)-dimensional faces of the resulting random polytope is an increasing function of \( n\). As a corollary, we show that the expected number of \( k\)-faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope.
MSC:
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 60D05 | Geometric probability and stochastic geometry |
| 52B11 | \(n\)-dimensional polytopes |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |
Keywords:
convex hull; Gaussian polytope; Gaussian zonotope; Goodman-Pollack model; \(f\)-vector; random polytope; regular polytopeReferences:
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