Reverse and dual Loomis-Whitney-type inequalities
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- by Stefano Campi, Richard J. Gardner and Paolo Gronchi PDF
- Trans. Amer. Math. Soc. 368 (2016), 5093-5124 Request permission
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.References
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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
- MR Author ID: 205850
- Email: campi@dii.unisi.it
- Richard J. Gardner
- Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
- MR Author ID: 195745
- 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
- MR Author ID: 340283
- Email: paolo@fi.iac.cnr.it
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 5093-5124
- MSC (2010): Primary 52A20, 52A40; Secondary 52A38
- DOI: https://doi.org/10.1090/tran/6668
- MathSciNet review: 3456173