Characterization of simplices via the Bezout inequality for mixed volumes
HTML articles powered by AMS MathViewer
- by Christos Saroglou, Ivan Soprunov and Artem Zvavitch PDF
- Proc. Amer. Math. Soc. 144 (2016), 5333-5340 Request permission
Abstract:
We consider the following Bezout inequality for mixed volumes: \[ V(K_1,\dots ,K_r,\Delta [{n-r}])V_n(\Delta )^{r-1} \leq \prod _{i=1}^r V(K_i,\Delta [{n-1}])\ \text { for }2\leq r\leq n.\] It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots , K_r$ in $\mathbb {R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots , K_r$ in $\mathbb {R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots , K_r$, then the boundary of $\Delta$ cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.References
- D. N. Bernstein, The number of roots of a system of equations, Funkcional. Anal. i Priložen. 9 (1975), no. 3, 1–4 (Russian). MR 0435072
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61 (Russian). MR 487230
- A. G. Kushnirenko, Newton polyhedra and Bezout’s theorem, (Russian) Funkcional. Anal. i Prilozhen. 10, no. 3, (1976) 82–83.
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- I. Soprunov and A. Zvavitch, Bezout Inequality for Mixed volumes, arXiv:1507.00765 [math.MG], International Mathematics Research Notices, to appear.
Additional Information
- Christos Saroglou
- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240
- MR Author ID: 915316
- ORCID: 0000-0001-5471-5560
- Email: csaroglo@math.kent.edu
- Ivan Soprunov
- Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
- MR Author ID: 672474
- Email: i.soprunov@csuohio.edu
- Artem Zvavitch
- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240
- MR Author ID: 671170
- Email: zvavitch@math.kent.edu
- Received by editor(s): December 16, 2015
- Received by editor(s) in revised form: February 11, 2016
- Published electronically: June 10, 2016
- Additional Notes: The third author was supported in part by U.S. National Science Foundation Grant DMS-1101636 and by the Simons Foundation.
- Communicated by: Thomas Schlumprecht
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5333-5340
- MSC (2010): Primary 52A39, 52A40, 52B11
- DOI: https://doi.org/10.1090/proc/13149
- MathSciNet review: 3556275