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On the reverse Lp–busemann–petty centroid inequality

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Stefano Campi  and
Paolo Gronchi
Stefano Campi
Affiliation:
Dipartimento di Matematica Pura e Applicata “G. Vitali”, Università degli Studi di Modena e Reggio Emilia, Via Campi 213 B, 41100 Modena, Italy. E-mail: [email protected]
Paolo Gronchi
Affiliation:
Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Via Madonna del Piano- Edificio F, 50019 Sesto Fiorentino (FI), Italy. E-mail: [email protected]
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Abstract

The volume of the Lp-centroid body of a convex body K ⊂ ℝd is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 1 - 11
Copyright
Copyright © University College London 2002

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References

[B]Blaschke, W.. Affine Geometric XIV: Eine Minimumaufgabc für Legendres Trägheitsellipsoid, Ber. Verh. Sächs. Akad. Leipzig, Math.-Phys. Ki. 70 (1918), 72–75.Google Scholar
[BB]Bisztriczky, T. and Böröczky, K. Jr., About the centroid body and the ellipsoid of inertia. Mathematika, to appear.Google Scholar
[CCG]Campi, S.. Colesanti, A. and Gronchi, P., A note on Sylvester's problem for random polytopcs in a convex body. Rend. 1st. Mat. Univ. Trieste 31 (1999), 79–94.Google Scholar
[CG]Campi, S. and Gronchi, P., The Lp-Busemann-Petty centroid inequality body, Adv. Math. 167 (2002), 128–141.Google Scholar
[Fa]Fáry, I.. Sur la dénsit des réscaux de domaines convexes, Bull. Soc. Math. France 78 (1950). 152–161.Google Scholar
[FR]Fáry, I. and Rédei, L.. Der zentralsymmetrische Kern und die zentralsymmetrische Hüllc von konvexen Körpern, Math. Ann. 122 (1950), 205–220.CrossRefGoogle Scholar
[Fi]Firey, W. J., p-means of convex bodies. Math. Scand. 10 (1962), 17–24.CrossRefGoogle Scholar
[G]Gardner, R. J., Geometric Tomography, Cambridge University Press, Cambridge, 1995.Google Scholar
[J1]John, F.. Polar correspondence with respect to convex regions, Duke Math. J. 3 (1937), 355–369.CrossRefGoogle Scholar
[J2]John, F.. Extremum problems with inequalities as subsidiary conditions, In Courant Anniversary Volume (Interscience, New York), 1948, 187–204.Google Scholar
[LM]Lindenstrauss, J. and Milman, V. D., Local theory of normed spaces and convexity, Handbook of Convex Geometry (Gruber, P. M. and Wills, J. M., eds.), North-Holland, Amsterdam, 1993, pp. 1149–1220.CrossRefGoogle Scholar
[L]Lutwak, E., The Brunn-Minkowski Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150.CrossRefGoogle Scholar
[LYZ]Lutwak, E., Yang, D. and Zhang, G., Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111–132.CrossRefGoogle Scholar
[LZ]Lutwak, E. and Zhang, G., Blaschke Santaló inequalities, J. Differential Geom. 47 (1997), 1–16.Google Scholar
[MP]Milman, V. D. and Pajor, A.. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geometric Aspects of Functional Analysis (Lindenstrauss, J. and Milman, V. D., eds.), vol. 1376, Springer Lecture Notes in Math., 1989, 64–104.CrossRefGoogle Scholar
[P1]Petty, C. M.. Centroid surfaces. Pacific J. Math. 11 (1961), 1535–1547.CrossRefGoogle Scholar
[P2]Petty, C. M., Ellipsoids, Convexity and its Applications (Gruber, P. M. and Wills, J. M., eds.). Birkhäuser, Basel, 1983, pp. 264–276.Google Scholar
[RS]Rogers, C. A. and Shephard, G. C., Some extremal problems for convex bodies, Mathematika 5 (1958), 93–102.CrossRefGoogle Scholar
[RT]Rogers, C. A. and Taylor, S. J., The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273–302.CrossRefGoogle Scholar
[Sc]Scheck, F., Mechanics, Springer-Verlag, Berlin Heidelberg, 1990.CrossRefGoogle Scholar
[Sc]Schneider, R., Convex bodies: the Brunn–Minkowski Theory, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[Sh]Shephard, G. C., Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229–36.CrossRefGoogle Scholar