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Mixed volumes and the Bochner method


Authors: Yair Shenfeld and Ramon van Handel
Journal: Proc. Amer. Math. Soc. 147 (2019), 5385-5402
MSC (2010): Primary 52A39, 52A40, 58J50
DOI: https://doi.org/10.1090/proc/14651
Published electronically: June 10, 2019
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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''.


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Additional Information

Yair Shenfeld
Affiliation: Sherrerd Hall 323, Princeton University, Princeton, New Jersey 08544
Email: yairs@princeton.edu

Ramon van Handel
Affiliation: Fine Hall 207, Princeton University, Princeton, New Jersey 08544
Email: rvan@princeton.edu

DOI: https://doi.org/10.1090/proc/14651
Keywords: Mixed volumes, mixed discriminants, Alexandrov-Fenchel inequality, Bochner method, hyperbolic quadratic forms, convex geometry
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
Article copyright: © Copyright 2019 American Mathematical Society