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Random ball-polyhedra and inequalities for intrinsic volumes

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Abstract

We prove a randomized version of the generalized Urysohn inequality relating mean width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.

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Acknowledgments

It is our pleasure to thank Rolf Schneider for helpful correspondence. We also thank Beatrice-Helen Vritsiou for helpful comments on a previous draft of this paper. Lastly, we thank the anonymous referee for careful reading and comments which improved the results and presentation of the paper.

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Authors and Affiliations

  1. Texas A&M University, College Station, USA

    Grigoris Paouris

  2. University of Missouri, Columbia, USA

    Peter Pivovarov

Authors
  1. Grigoris Paouris
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  2. Peter Pivovarov
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Correspondence to Peter Pivovarov.

Additional information

Communicated by A. Constantin.

Grigoris Paouris is supported by the A. Sloan Foundation, US NSF Grant CAREER-1151711 and BSF Grant 2010288. This work was partially supported by a Grant from the Simons Foundation (#317733 to Peter Pivovarov).

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Paouris, G., Pivovarov, P. Random ball-polyhedra and inequalities for intrinsic volumes. Monatsh Math 182, 709–729 (2017). https://doi.org/10.1007/s00605-016-0961-6

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