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Random projections of regular simplices. (English) Zbl 0751.52002

For a polytope \(P\subset\mathbb{R}^ n\) and a random \(d\)-dimensional subspace \(\Lambda\) in \(\mathbb{R}^ n\) with isotropic distribution let \(\Pi_ dP\) denote the orthogonal projection of \(P\) onto \(\Lambda\) and \(f_ k(\Pi_ dP)\) the number of \(k\)-faces of \(\Pi_ dP\). The authors present a formula for the expectation \(\mathbb{E} f_ k(\Pi_ dP)\) involving internal and external angles of \(P\). They then derive an asymptotic expression for \(\mathbb{E} f_ k(\Pi_ dP)\) (as \(n\) tends to infinity) in the case \(P\) is an \(n\)-dimensional regular simplex.
Reviewer: W.Weil (Karlsruhe)

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
60D05 Geometric probability and stochastic geometry
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References:

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