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INEQUALITIES BETWEEN MIXED VOLUMES OF CONVEX BODIES: VOLUME BOUNDS FOR THE MINKOWSKI SUM
Author(s) -
Averkov Gennadiy,
Borger Christopher,
Soprunov Ivan
Publication year - 2020
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/mtk.12055
Subject(s) - mixed volume , mathematics , convex body , minkowski space , upper and lower bounds , bounded function , combinatorics , minkowski addition , polytope , regular polygon , volume (thermodynamics) , tuple , support function , lattice (music) , discrete mathematics , convex set , mathematical analysis , convex optimization , geometry , physics , quantum mechanics , acoustics
Abstract In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sumP 1 + ⋯ + P dof d ‐dimensional lattice polytopes is bounded from above by a function of order O ( m 2 d ) , where m is the mixed volume of the tuple ( P 1 , ⋯ , P d ) . This is a consequence of the well‐known Aleksandrov–Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to O ( m d ) , which is asymptotically sharp. We furthermore prove a sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples of arbitrary convex bodies with volume at least one. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.