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Volume Inequalities for Lp-Zonotopes

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Stefano Campi  and
Paolo Gronchi
Stefano Campi
Affiliation:
Dipartimento di Ingegneria dell'Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy. E-mail: campi@dii.unisi.it
Paolo Gronchi
Affiliation:
Dipartimento di Matematica e Applicazioni per l'Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy. E-mail: paolo@fi.iac.cnr.it
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Abstract

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of ℓq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 71 - 80
Copyright
Copyright © University College London 2006

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References

1Ball, K. M., Mahler's conjecture and wavelets, Discrete Comput. Geom. 13 (1995), 271–277.CrossRefGoogle Scholar
2Ball, K. M., Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), 891–901.CrossRefGoogle Scholar
3Ball, K. M., Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), 351–359.CrossRefGoogle Scholar
4Barthe, F., On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335–361.CrossRefGoogle Scholar
5Barthe, F., A continuous version of the Brascamp-Lieb inequalities, in Geometric Aspects of Functional Analysis (2002–03), Lecture Notes in Mathematics 1850, Springer (Berlin, 2004), 53–64.CrossRefGoogle Scholar
6Bourgain, J. and Milman, V., New volume ratio properties for convex symmetric bodies in ℝn, Invent. Math. 88 (1987), 319–340.CrossRefGoogle Scholar
7Campi, S. and Gronchi, P., The Lp-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), 128–141.CrossRefGoogle Scholar
8Campi, S. and Gronchi, P., On volume product inequalities for convex sets, Proc. Amer. Math. Soc. 134 (2006), 2393–2402.CrossRefGoogle Scholar
9Campi, S. and Gronchi, P., Extremal convex sets for Sylvester-Busemann type functionals, Appl. Anal. 85 (2006), 129–141.CrossRefGoogle Scholar
10Firey, W. J., p-means of convex bodies, Math. Scand. 10 (1962), 17–24.CrossRefGoogle Scholar
11Gordon, Y., Meyer, M. and Reisner, S., Zonoids with minimal volume-product – a new proof, Proc. Amer. Math. Soc. 104 (1988), 273–276.Google Scholar
12Lutwak, E., Selected affine isoperimetric inequalities, in Handbook of Convex Geometry (eds. Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 151–176.CrossRefGoogle Scholar
13Lutwak, E., Yang, D. and Zhang, G., Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111–132.CrossRefGoogle Scholar
14Lutwak, E., Yang, D. and Zhang, G., Volume inequalities for subspaces of Lp, J. Differential Geom. 68 (2004), 159–184.CrossRefGoogle Scholar
15Lutwak, E., Yang, D. and Zhang, G., Volume inequalities for isotropic measures, Amer. J. Math. (to appear).Google Scholar
16Lutwak, E., Yang, D. and Zhang, G., Optimal Sobolev norms and the Lp-Minkowski problem, Int. Math. Res. Not. (2006), 62987, 1–21.Google Scholar
17Mahler, K., Ein Übertragungsprinzip für konvexe Körper, Casopis Pĕst. Mat. Fys. 68 (1939), 93–102.CrossRefGoogle Scholar
18Mahler, K., Ein Minimalproblem fü konvexe Polygone, Mathematica (Zutphen) B 7 (1939), 118–127.Google Scholar
19Meyer, M. and Pajor, A., On the Blaschke-Santaló inequality, Arch. Math. 55 (1990), 82–93.CrossRefGoogle Scholar
20Reisner, S., Random polytopes and the volume product of symmetric convex bodies, Math. Scand. 57 (1985), 386–392.CrossRefGoogle Scholar
21Reisner, S., Zonoids with minimal volume product, Math. Z. 192 (1986), 339–346.CrossRefGoogle Scholar
22Rogers, C. A. and Shephard, G. C., Some extremal problems for convex bodies, Mathematika 5 (1958), 93–102.CrossRefGoogle Scholar
23Schneider, R., Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
24Schneider, R. and Weil, W., Zonoids and related topics, in Convexity and its Applications (eds Gruber, P. M. and Wills, J. M.), Birkhäuser (Basel, 1983), 296–317.CrossRefGoogle Scholar
25Shephard, G. C., Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229–236.CrossRefGoogle Scholar