Reverse and dual Loomis-Whitney-type inequalities

Campi, Stefano; Gardner, Richard; Gronchi, Paolo

doi:10.1090/tran/6668

Reverse and dual Loomis-Whitney-type inequalities
HTML articles powered by AMS MathViewer

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

Paul Balister and Béla Bollobás, Projections, entropy and sumsets, Combinatorica 32 (2012), no. 2, 125–141. MR 2927635, DOI 10.1007/s00493-012-2453-1
Adi Ben-Israel, The change-of-variables formula using matrix volume, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 300–312. MR 1718666, DOI 10.1137/S0895479895296896
U. Betke and P. McMullen, Estimating the sizes of convex bodies from projections, J. London Math. Soc. (2) 27 (1983), no. 3, 525–538. MR 697145, DOI 10.1112/jlms/s2-27.3.525
Jonathan Bennett, Anthony Carbery, and Terence Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261–302. MR 2275834, DOI 10.1007/s11511-006-0006-4
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
Stefano Campi, Andrea Colesanti, and Paolo Gronchi, Convex bodies with extremal volumes having prescribed brightness in finitely many directions, Geom. Dedicata 57 (1995), no. 2, 121–133. MR 1347320, DOI 10.1007/BF01264932
Stefano Campi and Paolo Gronchi, Estimates of Loomis-Whitney type for intrinsic volumes, Adv. in Appl. Math. 47 (2011), no. 3, 545–561. MR 2822201, DOI 10.1016/j.aam.2011.01.001
D. R. Conant and W. A. Beyer, Generalized Pythagorean theorem, Amer. Math. Monthly 81 (1974), 262–265. MR 338322, DOI 10.2307/2319528
S. R. Finch, Mean width of a regular cross-polytope, arXiv:1112.0499.
Wm. J. Firey, Pythagorean inequalities for convex bodies, Math. Scand. 8 (1960), 168–170. MR 125496, DOI 10.7146/math.scand.a-10605
Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886, DOI 10.1017/CBO9781107341029
D. J. H. Garling, Inequalities: a journey into linear analysis, Cambridge University Press, Cambridge, 2007. MR 2343341, DOI 10.1017/CBO9780511755217
Misha Gromov, Entropy and isoperimetry for linear and non-linear group actions, Groups Geom. Dyn. 2 (2008), no. 4, 499–593. MR 2442946, DOI 10.4171/GGD/48
Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
Katalin Gyarmati, Máté Matolcsi, and Imre Z. Ruzsa, A superadditivity and submultiplicativity property for cardinalities of sumsets, Combinatorica 30 (2010), no. 2, 163–174. MR 2676833, DOI 10.1007/s00493-010-2413-6
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775
Te Sun Han, Nonnegative entropy measures of multivariate symmetric correlations, Information and Control 36 (1978), no. 2, 133–156. MR 464499
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1959.
Daniel Hug and Rolf Schneider, Reverse inequalities for zonoids and their application, Adv. Math. 228 (2011), no. 5, 2634–2646. MR 2838052, DOI 10.1016/j.aim.2011.07.018
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc 55 (1949), 961–962. MR 0031538, DOI 10.1090/S0002-9904-1949-09320-5
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375–390. MR 1781476, DOI 10.1215/S0012-7094-00-10432-2
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Volume inequalities for isotropic measures, Amer. J. Math. 129 (2007), no. 6, 1711–1723. MR 2369894, DOI 10.1353/ajm.2007.0038
N. Madras, D. W. Sumners, and S. G. Whittington, Almost unknotted embeddings of graphs in $\Bbb Z^3$ and higher dimensional analogues, J. Knot Theory Ramifications 18 (2009), no. 8, 1031–1048. MR 2554333, DOI 10.1142/S0218216509007336
Mathieu Meyer, A volume inequality concerning sections of convex sets, Bull. London Math. Soc. 20 (1988), no. 2, 151–155. MR 924244, DOI 10.1112/blms/20.2.151
H. Q. Ngo, E. Porat, C. Ré, and A. Rudra, Worst-case optimal join algorithms, Proc. 31st Symp. on Principles of Database Systems, Association for Computing Machinery, NY (2012), pp. 37–48. arXiv:1203.1952v1.
Washek F. Pfeffer, The Riemann approach to integration, Cambridge Tracts in Mathematics, vol. 109, Cambridge University Press, Cambridge, 1993. Local geometric theory. MR 1268404
Franz E. Schuster and Manuel Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Differential Geom. 92 (2012), no. 2, 263–283. MR 2998673
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
G. C. Shephard, Shadow systems of convex sets, Israel J. Math. 2 (1964), 229–236. MR 179686, DOI 10.1007/BF02759738