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Comparing the M-position with some classical positions of convex bodies

Published online by Cambridge University Press:  21 October 2011

E. MARKESSINIS
Affiliation:
Department of Mathematics, University of Athens, Panepistimioupolis 157-84, Athens, Greece. e-mail: lefteris128@yahoo.gr
G. PAOURIS
Affiliation:
Department of Mathematics, Texas A & M University, College Station, TX 77843U.S.A. e-mail: grigoris@math.tamu.edu
CH. SAROGLOU
Affiliation:
Department of Mathematics, University of Crete, 714-09 Heraklion, Crete, Greece. e-mail: saroglou@math.uoc.gr

Abstract

The purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

[1]Ball, K. M.Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2) 44 (1991), 351359.CrossRefGoogle Scholar
[2]Ball, K. M.Shadows of convex bodies. Trans. Amer. Math. Soc. 327 (1991), no. 2, 891901.CrossRefGoogle Scholar
[3]Bourgain, J. On the distribution of polynomials on high dimensional convex sets. Lecture Notes in Mathematics 1469 (Springer, 1991), 127–137.Google Scholar
[4]Bourgain, J., Klartag, B. and Milman, V. D.Symmetrization and isotropic constants of convex bodies In Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics 1850 (2004), 101116.CrossRefGoogle Scholar
[5]Bourgain, J. and Milman, V. D.New volume ratio properties for convex symmetric bodies in ℝd. Invent. Math. 88 (1987), 319340.CrossRefGoogle Scholar
[6]Gardner, R. J.Geometric tomography, Encyclopedia of Mathematics and its Applications 58 (Cambridge University Press, 1995).Google Scholar
[7]Giannopoulos, A. Notes on isotropic convex bodies. Warsaw University Notes (2003).Google Scholar
[8]Giannopoulos, A. and Milman, V. D.Extremal problems and isotropic positions of convex bodies. Israel J. Math. 117 (2000), 2960.CrossRefGoogle Scholar
[9]Giannopoulos, A. and Milman, V. D. Euclidean structure in finite-dimensional normed spaces. Handbook of the Geometry of Banach Spaces (Johnson-Lindenstrauss eds.), Vol. 1 (2001), 707–779.Google Scholar
[10]Giannopoulos, A., Paouris, G. and Valettas, P.On the existence of subgaussian directions for log-concave measures. Contemp. Math. 545 (2011), 103122.CrossRefGoogle Scholar
[11]Giannopoulos, A. and Papadimitrakis, M.Isotropic surface area measures. Mathematika 46 (1999), 113.CrossRefGoogle Scholar
[12]John, F.Extremum problems with inequalities as subsidiary conditions. Courant Anniversary Volume. (Interscience 1948), 187–204.Google Scholar
[13]Klartag, B.On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006), 12741290.CrossRefGoogle Scholar
[14]Klartag, B. and Milman, E. Centroid bodies and the logarithmic laplace transform - a unified approach. arXiv:1103.2985v1Google Scholar
[15]Milman, V. D.Inegalité de Brunn–Minkowski inverse et applications à la théorie locale des espaces normés. C.R. Acad. Sci. Paris 302 (1986), 2528.Google Scholar
[16]Milman, V. D.Isomorphic symmetrization and geometric inequalities. Geom. Aspects of Funct. Analysis (Lindnstrauss–Milman eds.). Lecture Notes in Math. 1317 (1988), 107131.CrossRefGoogle Scholar
[17]Milman, V. D. and Pajor, A.Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1376 (1989), 64104.CrossRefGoogle Scholar
[18]Milman, V. D. and Schechtman, G.Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200, (Springer, 1986).Google Scholar
[19]Petty, C. M.Surface area of a convex body under affine transformations. Proc. Amer. Math. Soc. 12 (1961), 824828.CrossRefGoogle Scholar
[20]Pisier, G.The volume of convex bodies and Banach space geometry. Cambridge Tracts in Math. 94 (1989).Google Scholar
[21]Saroglou, Ch. Minimal surface area position of a convex body is not always an M-position. Israel. J. Math. (to appear).Google Scholar
[22]Schneid, R.Convex bodies: the Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications 44 (Cambridge University Press, 1993).CrossRefGoogle Scholar
[23]Tomczak–Jaegermann, N.Banach–Mazur distances and finite dimensional operator ideals. Pitman Monographs 38 (Pitman, 1989).Google Scholar
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