Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T20:33:00.872Z Has data issue: false hasContentIssue false
  • English
  • Français

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: campi@unimo.it
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: paolo@fi.iac.cnr.it
Get access

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

Information

Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002, pp. 1 - 11
Copyright
Copyright © University College London 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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