Document Type
Article
Publication Date
10-2020
Publication Title
Mathematika
Abstract
n the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum P1++Pd of d-dimensional lattice polytopes is bounded from above by a function of order O(m2d), where m is the mixed volume of the tuple (P1,,Pd). 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(md), 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.
Repository Citation
Averkov, Gennadiy; Borger, Christopher; and Soprunov, Ivan, "Inequalities Between Mixed Volumes of Convex Bodies: Volume Bounds for the Minkowski Sum" (2020). Mathematics and Statistics Faculty Publications. 352.
https://engagedscholarship.csuohio.edu/scimath_facpub/352
DOI
10.1112/mtk.12055
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Volume
66
Issue
4
Comments
The two first authors and a research visit of the third author were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 314838170, GRK 2297 MathCoRe.