Abstract
In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum P 1 + ⋯ + P d of 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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
