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Discrete & Computational Geometry

, Volume 60, Issue 2, pp 406–419 | Cite as

A Center Transversal Theorem for an Improved Rado Depth

  • Pavle V. M. Blagojević
  • Roman Karasev
  • Alexander MagazinovEmail author
Article

Abstract

A celebrated result of Dol’nikov, and of Živaljević and Vrećica, asserts that for every collection of m measures \(\mu _1,\dots ,\mu _m\) on the Euclidean space \(\mathbb R^{n + m - 1}\) there exists a projection onto an n-dimensional vector subspace \(\Gamma \) with a point in it at depth at least \({1}/({n + 1})\) with respect to each associated n-dimensional marginal measure \(\Gamma _*\mu _1,\dots ,\Gamma _*\mu _m\). In this paper we consider a natural extension of this result and ask for a minimal dimension of a Euclidean space in which one can require that for any collection of m measures there exists a vector subspace \(\Gamma \) with a point in it at depth slightly greater than \({1}/({n + 1})\) with respect to each n-dimensional marginal measure. In particular, we prove that if the required depth is \({1}/({n + 1}) + {1}/({3(n + 1)^3})\) then the increase in the dimension of the ambient space is a linear function in both m and n.

Keywords

Tukey depth Rado theorem Centerline Center transversal 

Mathematics Subject Classification

52C35 52A35 68U05 

Notes

Acknowledgements

The authors are grateful to Vladimir Dol’nikov and Gaiane Panina for suggesting the several-measure setup, and to Bo’az Klartag for pointing out some of the needed constructions. Furthermore we want to express our gratitude to the referee for excellent observations and many useful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany
  2. 2.Matematički Institut SANUBeogradSerbia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.Institute for Information Transmission Problems RASMoscowRussia
  5. 5.School of MathematicsTel Aviv UniversityTel AvivIsrael

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