Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Characterization of simplices via the Bezout inequality for mixed volumes


Authors: Christos Saroglou, Ivan Soprunov and Artem Zvavitch
Journal: Proc. Amer. Math. Soc. 144 (2016), 5333-5340
MSC (2010): Primary 52A39, 52A40, 52B11
DOI: https://doi.org/10.1090/proc/13149
Published electronically: June 10, 2016
MathSciNet review: 3556275
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the following Bezout inequality for mixed volumes:

$\displaystyle 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}])\ $$\displaystyle \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 [Enhancements On Off] (What's this?)

  • [B] 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
  • [F] 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
  • [Kh] A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61 (Russian). MR 487230
  • [Ku] A. G. Kushnirenko, Newton polyhedra and Bezout's theorem, (Russian) Funkcional. Anal. i Prilozhen. 10, no. 3, (1976) 82-83.
  • [Sch] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521
  • [SZ] I. Soprunov and A. Zvavitch, Bezout Inequality for Mixed volumes, arXiv:1507.00765 [math.MG], International Mathematics Research Notices, to appear.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52A39, 52A40, 52B11

Retrieve articles in all journals with MSC (2010): 52A39, 52A40, 52B11


Additional Information

Christos Saroglou
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240
Email: csaroglo@math.kent.edu

Ivan Soprunov
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
Email: i.soprunov@csuohio.edu

Artem Zvavitch
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44240
Email: zvavitch@math.kent.edu

DOI: https://doi.org/10.1090/proc/13149
Keywords: Convex bodies, mixed volume, convex polytopes, Bezout inequality, Aleksandrov--Fenchel inequality
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
Article copyright: © Copyright 2016 American Mathematical Society