Mixed volumes and the Bochner method
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- by Yair Shenfeld and Ramon van Handel PDF
- Proc. Amer. Math. Soc. 147 (2019), 5385-5402 Request permission
Abstract:
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes many known inequalities as special cases. The aim of this note is to give new proofs of the Alexandrov-Fenchel inequality and of its matrix counterpart, Alexandrov’s inequality for mixed discriminants, that appear conceptually and technically simpler than earlier proofs and clarify the underlying structure. Our main observation is that these inequalities can be reduced by the spectral theorem to certain trivial “Bochner formulas”.References
- A. D. Alexandrov. Zur Theorie der gemischten Volumina von konvexen Körpern II. Mat. Sbornik N.S., 2:1205–1238, 1937.
- A. D. Alexandrov. Zur Theorie der gemischten Volumina von konvexen Körpern IV. Mat. Sbornik N.S., 3:227–251, 1938.
- S. Artstein-Avidan, D. Florentin, and Y. Ostrover, Remarks about mixed discriminants and volumes, Commun. Contemp. Math. 16 (2014), no. 2, 1350031, 14. MR 3195153, DOI 10.1142/S0219199713500314
- Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, Asymptotic geometric analysis. Part I, Mathematical Surveys and Monographs, vol. 202, American Mathematical Society, Providence, RI, 2015. MR 3331351, DOI 10.1090/surv/202
- R. B. Bapat and T. E. S. Raghavan, Nonnegative matrices and applications, Encyclopedia of Mathematics and its Applications, vol. 64, Cambridge University Press, Cambridge, 1997. MR 1449393, DOI 10.1017/CBO9780511529979
- T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR 920366
- Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York-Berlin, 1983. MR 683612, DOI 10.1007/978-1-4612-1148-8
- 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
- W. Fenchel, Inégalités quadratiques entre les volumes mixtes des corps convexes, C. R. Acad. Sci. Paris 203 (1936), 647–650.
- Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 3rd ed., Universitext, Springer-Verlag, Berlin, 2004. MR 2088027, DOI 10.1007/978-3-642-18855-8
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
- M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1–38. MR 1095529
- David Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea Publishing Co., New York, N.Y., 1953 (German). MR 0056184
- André Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III, Dunod, Paris, 1958 (French). MR 0124009
- H. Minkowski, Gesammelte Abhandlungen. Zweiter Band, B.G. Teubner, 1911.
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
- Xu Wang, A remark on the Alexandrov-Fenchel inequality, J. Funct. Anal. 274 (2018), no. 7, 2061–2088. MR 3762095, DOI 10.1016/j.jfa.2018.01.016
Additional Information
- Yair Shenfeld
- Affiliation: Sherrerd Hall 323, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 1271383
- Email: yairs@princeton.edu
- Ramon van Handel
- Affiliation: Fine Hall 207, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 761136
- Email: rvan@princeton.edu
- Received by editor(s): November 21, 2018
- Received by editor(s) in revised form: February 19, 2019, and March 8, 2019
- Published electronically: June 10, 2019
- Additional Notes: This work was supported in part by NSF grants CAREER-DMS-1148711 and DMS-1811735, ARO through PECASE award W911NF-14-1-0094, and the Simons Collaboration on Algorithms & Geometry. This work was initiated while the authors were in residence at MSRI in Berkeley, CA, supported by NSF grant DMS-1440140. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis is gratefully acknowledged.
- Communicated by: Deane Yang
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5385-5402
- MSC (2010): Primary 52A39, 52A40, 58J50
- DOI: https://doi.org/10.1090/proc/14651
- MathSciNet review: 4021097