Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
-
Silouanos Brazitikos
and
Dimitris-Marios Liakopoulos
Abstract
The classical Loomis–Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer’s dual Loomis–Whitney inequality.
References
[1] S. Artstein-Avidan, D. Florentin, Y. Ostrover, Remarks about mixed discriminants and volumes. Commun. Contemp. Math. 16 (2014), 1350031, 14 pp. MR3195153 Zbl 1292.5200410.1142/S0219199713500314Search in Google Scholar
[2] S. Artstein-Avidan, A. Giannopoulos, V. D. Milman, Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2015. MR3331351 Zbl 1337.5200110.1090/surv/202Search in Google Scholar
[3] K. Ball, Shadows of convex bodies. Trans. Amer. Math. Soc. 327 (1991), 891–901. MR1035998 Zbl 0746.5200710.1090/S0002-9947-1991-1035998-3Search in Google Scholar
[4] K. Ball, Convex geometry and functional analysis. In: Handbook of the geometry of Banach spaces, Vol. I, 161–194, North-Holland 2001. MR1863692 Zbl 1017.4600410.1016/S1874-5849(01)80006-1Search in Google Scholar
[5] F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998), 335–361. MR1650312 Zbl 0901.2601010.1007/s002220050267Search in Google Scholar
[6] L. Berwald, Verallgemeinerung eines Mittelwertsatzes von J. Favard für positive konkave Functionen. Acta Math. 79 (1947), 17-37. MR0021036 Zbl 0029.1170410.1007/BF02404692Search in Google Scholar
[7] U. Betke, W. Weil, Isoperimetric inequalities for the mixed area of plane convex sets. Arch. Math. (Basel)57 (1991), 501–507. MR1129527 Zbl 0765.5201110.1007/BF01246750Search in Google Scholar
[8] B. Bollobás, A. Thomason, Projections of bodies and hereditary properties of hypergraphs. Bull. London Math. Soc. 27 (1995), 417–424. MR1338683 Zbl 0836.0507210.1112/blms/27.5.417Search in Google Scholar
[9] S. Brazitikos, A. Giannopoulos, P. Valettas, B.-H. Vritsiou, Geometry of isotropic convex bodies, volume 196 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2014. MR3185453 Zbl 1304.5200110.1090/surv/196Search in Google Scholar
[10] Y. D. Burago, V. A. Zalgaller, Geometric inequalities. Springer 1988. MR936419 Zbl 0633.5300210.1007/978-3-662-07441-1Search in Google Scholar
[11] A. Dembo, T. M. Cover, J. A. Thomas, Information-theoretic inequalities. IEEE Trans. Inform. Theory37 (1991), 1501–1518. MR1134291 Zbl 0741.9400110.1109/18.104312Search in Google Scholar
[12] M. Fradelizi, A. Giannopoulos, M. Meyer, Some inequalities about mixed volumes. Israel J. Math. 135 (2003), 157–179. MR1997041 Zbl 1045.5200210.1007/BF02776055Search in Google Scholar
[13] R. J. Gardner, Geometric tomography, volume 58 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2006. MR2251886 Zbl 1102.5200210.1017/CBO9781107341029Search in Google Scholar
[14] A. Giannopoulos, M. Hartzoulaki, G. Paouris, On a local version of the Aleksandrov–Fenchel inequality for the quermassintegrals of a convex body. Proc. Amer. Math. Soc. 130 (2002), 2403–2412. MR1897466 Zbl 1003.5200510.1090/S0002-9939-02-06329-3Search in Google Scholar
[15] A. Giannopoulos, A. Koldobsky, P. Valettas, Inequalities for the surface area of projections of convex bodies. Canad. J. Math., accepted, 23 pages.10.4153/CJM-2016-051-xSearch in Google Scholar
[16] D. Hug, R. Schneider, Reverse inequalities for zonoids and their application. Adv. Math. 228 (2011), 2634–2646. MR2838052 Zbl 1230.5202110.1016/j.aim.2011.07.018Search in Google Scholar
[17] A.-J. Li, Q. Huang, The dual Loomis-Whitney inequality. Bull. Lond. Math. Soc. 48 (2016), 676–690. MR3532142 Zbl 1351.5200610.1112/blms/bdw031Search in Google Scholar
[18] L. H. Loomis, H. Whitney, An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc55 (1949), 961–962. MR0031538 Zbl 0035.3830210.1007/978-1-4612-2974-2_34Search in Google Scholar
[19] M. Meyer, A volume inequality concerning sections of convex sets. Bull. London Math. Soc. 20 (1988), 151–155. MR924244 Zbl 0639.5200910.1112/blms/20.2.151Search in Google Scholar
[20] G. Paouris, Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), 1021–1049. MR2276533 Zbl 1114.5200410.1007/s00039-006-0584-5Search in Google Scholar
[21] G. Paouris, Small ball probability estimates for log-concave measures. Trans. Amer. Math. Soc. 364 (2012), 287–308. MR2833584 Zbl 1248.6002710.1090/S0002-9947-2011-05411-5Search in Google Scholar
[22] C. Saroglou, I. Soprunov, A. Zvavitch, Characterization of simplices via the Bezout inequality for mixed volumes.Proc. Amer. Math. Soc. 144 (2016), 5333–5340. MR3556275 Zbl 1351.5200510.1090/proc/13149Search in Google Scholar
[23] R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2014. MR3155183 Zbl 1287.52001Search in Google Scholar
[24] I. Soprunov, A. Zvavitch, Bezout inequality for mixed volumes. Int. Math. Res. Not. IMRN (2016), no. 23, 7230–7252. MR363208110.1093/imrn/rnv390Search in Google Scholar
© 2018 Walter de Gruyter GmbH Berlin/Boston
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
Articles in the same Issue
- Frontmatter
- Graphs and metric 2-step nilpotent Lie algebras
- Stein manifolds of nonnegative curvature
- Homogeneous spin Riemannian manifolds with the simplest Dirac operator
- On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
- Harmonicity of vector fields on a class of Lorentzian solvable Lie groups
- Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
- Criteria for Hattori–Masuda multi-polytopes via Duistermaat–Heckman functions and winding numbers
- On complex Berwald metrics which are not conformal changes of complex Minkowski metrics
- Continuous space-time transformations
