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VOLUME ESTIMATES FOR L p ‐ZONOTOPES AND BEST BEST CONSTANTS IN BRASCAMP–LIEB INEQUALITIES
Author(s) -
Alonso-Gutiérrez David
Publication year - 2010
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/s0025579309000345
Subject(s) - mathematics , volume (thermodynamics) , regular polygon , position (finance) , combinatorics , convex body , transformation (genetics) , constant (computer programming) , upper and lower bounds , span (engineering) , pure mathematics , mathematical analysis , geometry , convex optimization , chemistry , thermodynamics , physics , biochemistry , civil engineering , finance , gene , engineering , economics , computer science , programming language
Given some unit vectors a 1 ,…, a m ∈ℝ n that span all ℝ n and some positive numbers θ 1 ,…, θ m , we consider for every p ≥1 the convex bodyK p : = { x ∈ ℝ n : ∑ i = 1 m| 〈 x ,a iθ i〉 | p ≤ 1} .We will give some upper bounds for the volume of K p and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp–Lieb inequality. We will also see how the best choice of these parameters is related to the transformation which takes K p to a special position which, for instance, when p = ∞ , is John's position.