Sectional bodies associated with a convex body
HTML articles powered by AMS MathViewer
- by Matthieu Fradelizi
- Proc. Amer. Math. Soc. 128 (2000), 2735-2744
- DOI: https://doi.org/10.1090/S0002-9939-00-05342-9
- Published electronically: February 28, 2000
- PDF | Request permission
Abstract:
We define the sectional bodies associated to a convex body in $\mathbb {R}^n$ and two related measures of symmetry. These definitions extend those of Grünbaum (1963). As Grünbaum conjectured, we prove that the simplices are the most dissymmetrical convex bodies with respect to these measures. In the case when the convex body has a sufficiently smooth boundary, we investigate some limit behaviours of the volume of the sectional bodies.References
- M. Fradelizi, Sections of convex bodies through their centroid, Arch. Math. (Basel) 69 (1997), no. 6, 515–522. MR 1480519, DOI 10.1007/s000130050154
- Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- Branko Grünbaum, Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270. MR 0156259
- Y. Kovetz: Some extremal problems on convex bodies, Thesis of the Hebrew University, Jerusalem, 1962.
- Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151–176. MR 1242979, DOI 10.1016/B978-0-444-89596-7.50010-9
- Endre Makai Jr. and Horst Martini, On bodies associated with a given convex body, Canad. Math. Bull. 39 (1996), no. 4, 448–459. MR 1426690, DOI 10.4153/CMB-1996-053-7
- E. Makai Jr., H. Martini and T. Ódor: Maximal sections and centrally symmetric bodies. Preprint.
- M. Meyer and E. Werner: The Santaló regions of a convex body. To appear in Trans. Amer. Math. Soc.
- M. Schmuckenschläger, The distribution function of the convolution square of a convex symmetric body in $\textbf {R}^n$, Israel J. Math. 78 (1992), no. 2-3, 309–334. MR 1194970, DOI 10.1007/BF02808061
- 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
- Carsten Schütt and Elisabeth Werner, The convex floating body, Math. Scand. 66 (1990), no. 2, 275–290. MR 1075144, DOI 10.7146/math.scand.a-12311
- Elisabeth Werner, Illumination bodies and affine surface area, Studia Math. 110 (1994), no. 3, 257–269. MR 1292847, DOI 10.4064/sm-110-3-257-269
Bibliographic Information
- Matthieu Fradelizi
- Affiliation: Université de Marne-la-Vallée, Equipe d’Analyse et de Mathématiques Appliquées, Cité Descartes, 5 Bd Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France
- MR Author ID: 626525
- Email: fradeliz@math.univ-mlv.fr
- Received by editor(s): May 11, 1998
- Received by editor(s) in revised form: October 29, 1998
- Published electronically: February 28, 2000
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2735-2744
- MSC (2000): Primary 52A20, 52A40, 53A05, 53A15
- DOI: https://doi.org/10.1090/S0002-9939-00-05342-9
- MathSciNet review: 1664362