For an isotropic convex body K⊂Rn we consider the isotropic constant LK of the symmetric random polytope KN generated by N independent random points which are distributed according to the cone probability measure on the boundary of K. We show that with overwhelming probability LK≤Clog(2N∕n), where C∈(0,∞) is an absolute constant. If K is unconditional we argue that even LK≤C with overwhelming probability and thereby verify the hyperplane conjecture for this model. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new ψ2-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.

Prochno, J., Thale, C., Turchi, N. (2019). The isotropic constant of random polytopes with vertices on convex surfaces. JOURNAL OF COMPLEXITY, 54 [10.1016/j.jco.2019.01.001].

The isotropic constant of random polytopes with vertices on convex surfaces

Turchi N.
2019

Abstract

For an isotropic convex body K⊂Rn we consider the isotropic constant LK of the symmetric random polytope KN generated by N independent random points which are distributed according to the cone probability measure on the boundary of K. We show that with overwhelming probability LK≤Clog(2N∕n), where C∈(0,∞) is an absolute constant. If K is unconditional we argue that even LK≤C with overwhelming probability and thereby verify the hyperplane conjecture for this model. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new ψ2-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.
Articolo in rivista - Articolo scientifico
Asymptotic geometric analysis; Bernstein inequality; Convex body; Hyperplane conjecture; Isotropic constant; Random polytope;
English
Prochno, J., Thale, C., Turchi, N. (2019). The isotropic constant of random polytopes with vertices on convex surfaces. JOURNAL OF COMPLEXITY, 54 [10.1016/j.jco.2019.01.001].
Prochno, J; Thale, C; Turchi, N
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/395180
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