Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-28T15:41:10.456Z Has data issue: false hasContentIssue false
  • English
  • Français

Shadow Systems and Volumes of Polar Convex Bodies

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Mathieu Meyer  and
Shlomo Reisner
Mathieu Meyer
Affiliation:
Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Cité Descartes, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France. E-mail: mathieu.meyer@univ-mlv.fr
Shlomo Reisner
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel. E-mail: reisner@math.haifa.ac.il
Get access

Abstract

It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t, thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝd that have at most d + 3 vertices.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 129 - 148
Copyright
Copyright © University College London 2006

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.)

References

1Ball, K. M., Logarithmically concave functions and sections of convex sets in ℝn. Studia Math. 88 (1988), 68–84.CrossRefGoogle Scholar
2Blaschke, W., Kreis und Kugel. Göschen (Leipzig, 1916); reproduced by Chelsea (New York, 1949).Google Scholar
3Bourbaki, N., Fonctions d'une Variable Réelle: Théorie Élémentaire. Hermann (Paris, 1976) and Nicolas Bourbaki (1982).Google Scholar
4Bourgain, J. and Milman, V. D., New volume ratio properties for convex symmetric bodies in ℝn. Invent. Math. 88 (1987), 319–340.CrossRefGoogle Scholar
5Busemann, H., A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 27–31.CrossRefGoogle ScholarPubMed
6Campi, S. and Gronchi, P., On volume product inequalities for convex sets. Proc. Amer. Math. Soc. 134 (2006), 2393–2402.CrossRefGoogle Scholar
7Gordon, Y., Meyer, M. and Reisner, S., Zonoids with minimal volume product – a new proof. Proc. Amer. Math. Soc. 104 (1988), 273–276.Google Scholar
8Lutwak, E. and Zhang, G., Blaschke-Santaló inequalities. J. Diff. Geom. 47 (1997), 1–16.Google Scholar
9Mahler, K., Ein Minimalproblem für konvexe Polygone. Mathematica (Zutphen) B 7 (1939), 118–127.Google Scholar
10Meyer, M., Une caractérisation volumique de certains espaces normés. Israel J. Math. 55 (1986), 317–326.CrossRefGoogle Scholar
11Meyer, M. and Pajor, A., On the Blaschke-Santaló inequality. Arch. Math. 55 (1990), 82–93.CrossRefGoogle Scholar
12Meyer, M., Convex bodies with minimal volume product in ℝ2. Monatsh. Math. 112 (1991), 297–301.CrossRefGoogle Scholar
13Meyer, M. and Reisner, S., Inequalities involving integrals of polar-conjugate concave functions. Monatsh. Math. 125 (1998), 219–227.CrossRefGoogle Scholar
14Petty, C. M., Affine isoperimetric problems. Ann. New York Acad. Sci. 440 (1985), 113–127.CrossRefGoogle Scholar
15Reisner, S., Zonoids with minimal volume product. Math. Zeitsch. 192 (1986), 339–346.Google Scholar
16Reisner, S., Minimal volume product in Banach spaces with a l-unconditional basis. J. London Math. Soc. 36 (1987), 126–136.CrossRefGoogle Scholar
17Rogers, C. A. and Shephard, G. C., Some extremal problems for convex bodies. Mathematika 5 (1958), 93–102.Google Scholar
18Saint-Raymond, J., Sur le volume des corps convexes symétriques. Séminaire d'Initiation à l'Analyse 1980–1981, Université Paris VI (Paris, 1981).Google Scholar
19Santaló, L. A., Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugal. Math. 8 (1949), 155–161.Google Scholar
20Schneider, R., Convex Bodies: the Brunn-Minkowski Theory (Encyclopedia of Mathematics and its Applications 44). Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
21Shephard, G. C., Shadow systems of convex bodies. Israel J. Math. 2 (1964), 229–236.Google Scholar