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On approximation by projections of polytopes with few facets

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Abstract

We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a section of an N-dimensional simplex one has

$$d(B,K) \geqslant c\sqrt {\frac{n}{{\ln \frac{{2N\ln (2N)}}{n}}}} $$

, where d(·, ·) denotes the Banach-Mazur distance and c is an absolute positive constant. The result is sharp up to a logarithmic factor.

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Authors and Affiliations

  1. Depatment of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

    Alexander E. Litvak

  2. Department of Mathematics, University of Michigan East Hall, 530 Church Street, Ann Arbor, Michigan, 48109, USA

    Mark Rudelson

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

    Nicole Tomczak-Jaegermann

Authors
  1. Alexander E. Litvak
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  2. Mark Rudelson
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  3. Nicole Tomczak-Jaegermann
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Correspondence to Alexander E. Litvak.

Additional information

In memoriam of Joram Lindenstrauss, a teacher and a colleague

Research partially supported by the E. W. R. Steacie Memorial Fellowship.

Research partially supported by NSF grant DMS 1161372.

This author holds the Canada Research Chair in Geometric Analysis.

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Litvak, A.E., Rudelson, M. & Tomczak-Jaegermann, N. On approximation by projections of polytopes with few facets. Isr. J. Math. 203, 141–160 (2014). https://doi.org/10.1007/s11856-014-0017-3

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  • DOI: https://doi.org/10.1007/s11856-014-0017-3

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