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On the Isotropic Constant of Random Polytopes

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Abstract

Let \(X_1,\ldots ,X_N\) be independent random vectors uniformly distributed on an isotropic convex body \(K\subset \mathbb {R}^n\), and let \(K_N\) be the symmetric convex hull of \(X_i\)’s. We show that with high probability \( L_{K_N}\le C \sqrt{\log (2N/n)}\), where \(C\) is an absolute constant. This result closes the gap in known estimates in the range \(Cn\le N\le n^{1+\delta }\). Furthermore, we extend our estimates to the symmetric convex hulls of vectors \(y_1 X_1, \dots , y_N X_N\), where \(y=(y_1, \dots , y_N)\) is a vector in \(\mathbb {R}^N\). Finally, we discuss the case of a random vector \(y\).

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Acknowledgments

Part of this work was done while the first named author was a postdoctoral fellow at the Department of Mathematical and Statistical Sciences at University of Alberta, and continued while he was a “Juan de la Cierva” postdoctoral researcher at the Mathematics Department in Universidad de Murcia. He would like to thank both departments for providing support, good environment and working conditions. We would also like to thank the anonymous referee for careful reading of the manuscript and for informing us that Theorem 1.1 has subsequently been proved also in [17] using essentially the same approach. Finally we are very thankful to Rafał Latała for showing us the argument in Lemma 5.2. David Alonso-Gutiérrez is Partially supported by MICINN project MTM2010-16679, MICINN-FEDER project MTM2009-10418, “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 04540/GERM/06 and Institut Universitari de Matemàtiques i Aplicacions de Castelló Alexander E. Litvak Research partially supported by the E.W.R. Steacie Memorial Fellowship Nicole Tomczak-Jaegermann holds the Canada Research Chair in Geometric Analysis.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Jaume I, Campus de Riu Sec, 12071, Castelló de la Plana, Spain

    David Alonso-Gutiérrez

  2. Department of Mathematics and Statistics Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Alexander E. Litvak & Nicole Tomczak-Jaegermann

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  1. David Alonso-Gutiérrez
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  2. Alexander E. Litvak
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  3. Nicole Tomczak-Jaegermann
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Correspondence to Alexander E. Litvak.

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Alonso-Gutiérrez, D., Litvak, A.E. & Tomczak-Jaegermann, N. On the Isotropic Constant of Random Polytopes. J Geom Anal 26, 645–662 (2016). https://doi.org/10.1007/s12220-015-9567-9

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  • DOI: https://doi.org/10.1007/s12220-015-9567-9

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