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Bulletin of the London Mathematical Society Advance Access originally published online on August 13, 2009
Bulletin of the London Mathematical Society 2009 41(5):853-858; doi:10.1112/blms/bdp062
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© 2009 London Mathematical Society

A sharp combinatorial version of Vaaler's theorem

K. M. Ball

Department of Mathematics
University College London
Gower Street
London
WC1E 6BT
United Kingdom
kmb@math.ucl.ac.uk

M. Prodromou

Department of Mathematics
University College London
Gower Street
London
WC1E 6BT
United Kingdom

Received 12 February 2008. Revision received 13 January 2009.

In 1979 Vaaler proved that every d-dimensional central section of the cube [–1, 1]n has volume at least 2d. We prove the following sharp combinatorial analogue. Let K be a d-dimensional subspace of Rn. Then, there exists a probability measure P on the section [–1, 1]n {cap} K such that the quadratic form


Formula

dominates the identity on K (in the sense that the difference is positive semi-definite).

2000 Mathematics Subject Classification 52A40 (primary), 52C35 (secondary).

The first author is partially supported by EPSRC Grant EP/E00296X/1. The second author is supported by EPSRC Grant EP/E00296X/1.


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