On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body
HTML articles powered by AMS MathViewer
- by A. Giannopoulos, M. Hartzoulaki and G. Paouris PDF
- Proc. Amer. Math. Soc. 130 (2002), 2403-2412 Request permission
Abstract:
We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality $W_i^2\geq W_{i-1}W_{i+1}$ which relates the quermassintegrals of a convex body $K$ to those of an arbitrary hyperplane projection of $K$. A consequence is the following fact: for any convex body $K$, for any $(n-1)$-dimensional subspace $E$ of ${\mathbb R}^n$ and any $t>0$, \begin{equation*}\frac {|P_E(K)+tD_E|}{|P_E(K)|}\leq \frac {|K+2tD_n|}{|K|},\end{equation*} where $D$ denotes the Euclidean unit ball and $|\cdot |$ denotes volume in the appropriate dimension.References
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Edwin F. Beckenbach and Richard Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 30, Springer-Verlag, Inc., New York, 1965. Second revised printing. MR 0192009
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
- Amir Dembo, Thomas M. Cover, and Joy A. Thomas, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1501–1518. MR 1134291, DOI 10.1109/18.104312
- M. Fradelizi, A. Giannopoulos and M. Meyer, Some inequalities about mixed volumes, preprint.
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- 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
- Rolf Schneider, On the Aleksandrov-Fenchel inequality, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 132–141. MR 809200, DOI 10.1111/j.1749-6632.1985.tb14547.x
Additional Information
- A. Giannopoulos
- Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
- Email: apostolo@math.uch.gr
- M. Hartzoulaki
- Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
- Email: hmarian@math.uch.gr
- G. Paouris
- Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
- MR Author ID: 671202
- Email: paouris@math.uch.gr
- Received by editor(s): December 20, 2000
- Received by editor(s) in revised form: March 16, 2001
- Published electronically: January 23, 2002
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2403-2412
- MSC (1991): Primary 52A20; Secondary 52A39, 52A40
- DOI: https://doi.org/10.1090/S0002-9939-02-06329-3
- MathSciNet review: 1897466