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BRASCAMP–LIEB INEQUALITY AND QUANTITATIVE VERSIONS OF HELLY'S THEOREM
Author(s) -
Brazitikos Silouanos
Publication year - 2016
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579316000255
Subject(s) - isoperimetric inequality , mathematics , complement (music) , inequality , constant (computer programming) , ball (mathematics) , pure mathematics , combinatorics , mathematical analysis , biochemistry , chemistry , complementation , computer science , programming language , gene , phenotype
We provide new quantitative versions of Helly's theorem. For example, we show that for every family { P i : i ∈ I } of closed half‐spaces in R n such that P = ⋂ i ∈ IP ihas positive volume, there exist s ⩽ α n andi 1 , … , i s ∈ I such thatvol n ( P i 1∩ ⋯ ∩ P i s) ⩽ ( C n ) nvol n ( P ) ,where α , C > 0 are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.