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Reverse and dual Loomis-Whitney-type inequalities


Authors: Stefano Campi, Richard J. Gardner and Paolo Gronchi
Journal: Trans. Amer. Math. Soc. 368 (2016), 5093-5124
MSC (2010): Primary 52A20, 52A40; Secondary 52A38
DOI: https://doi.org/10.1090/tran/6668
Published electronically: October 20, 2015
MathSciNet review: 3456173
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Abstract: Various results are proved giving lower bounds for the $ m$th intrinsic volume $ V_m(K)$, $ m=1,\dots ,n-1$, of a compact convex set $ K$ in $ \mathbb{R}^n$, in terms of the $ m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $ m=1$ and $ m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $ V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983.


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Additional Information

Stefano Campi
Affiliation: Dipartimento di Ingegneria dell’Informazione e di Scienze Matematiche, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy
Email: campi@dii.unisi.it

Richard J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
Email: Richard.Gardner@wwu.edu

Paolo Gronchi
Affiliation: Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy
Email: paolo@fi.iac.cnr.it

DOI: https://doi.org/10.1090/tran/6668
Keywords: Convex body, zonoid, intrinsic volume, Loomis-Whitney inequality, Meyer's inequality, Betke-McMullen conjecture, Cauchy-Binet theorem, geometric tomography
Received by editor(s): June 25, 2014
Published electronically: October 20, 2015
Additional Notes: The second author was supported in part by U.S. National Science Foundation Grant DMS-1103612.
Article copyright: © Copyright 2015 American Mathematical Society