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A sharp combinatorial version of Vaaler's theorem
Author(s) -
Ball K. M.,
Prodromou M.
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp062
Subject(s) - mathematics , section (typography) , combinatorics , subspace topology , cube (algebra) , identity (music) , measure (data warehouse) , quadratic equation , dimension (graph theory) , discrete mathematics , volume (thermodynamics) , pure mathematics , mathematical analysis , geometry , quantum mechanics , physics , database , advertising , acoustics , computer science , business
In 1979 Vaaler proved that every d ‐dimensional central section of the cube [−1, 1] n has volume at least 2 d . We prove the following sharp combinatorial analogue. Let K be a d ‐dimensional subspace of ℝ n . Then, there exists a probability measure P on the section [−1, 1] n ∩ K such that the quadratic form∫[ − 1 , 1 ] n ∩ Kυ ⊗ υ d P ( υ )dominates the identity on K (in the sense that the difference is positive semi‐definite).