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Intrinsic volumes of inscribed random polytopes in smooth convex bodies

  • I. Bárány (a1), F. Fodor (a2) and V. Vígh (a3)

Abstract

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by K n the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes V s (K n ) of K n for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of K n . The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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Copyright

Corresponding author

Postal address: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK. Email address: barany@renyi.hu
∗∗ Postal address: Department of Geometry, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: fodorf@math.u-szeged.hu
∗∗∗ Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary. Email address: vigvik@math.u-szeged.hu

Footnotes

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Supported by Hungarian OTKA grant 60427.

Supported by Hungarian OTKA grants 68398 and 75016, and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

Supported by Hungarian OTKA grant 75016.

Footnotes

References

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Intrinsic volumes of inscribed random polytopes in smooth convex bodies

  • I. Bárány (a1), F. Fodor (a2) and V. Vígh (a3)

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