Upper bounds for the volume and diameter of -dimensional sections of convex bodies

Author:
Jesús Bastero

Journal:
Proc. Amer. Math. Soc. **135** (2007), 1851-1859

MSC (2000):
Primary 46B20, 52A20, 52A40

Published electronically:
January 5, 2007

MathSciNet review:
2286096

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Abstract: In this paper some upper bounds for the volume and diameter of central sections of symmetric convex bodies are obtained in terms of the isotropy constant of the polar body. The main consequence is that every symmetric convex body in of volume one has a proportional section , dim ( ), of diameter bounded by

**[B1]**K. BALL,*Isometric problems in and sections of convex bodies*, Ph.D. Thesis, Cambridge University (1986).**[B2]**Keith Ball,*Logarithmically concave functions and sections of convex sets in 𝑅ⁿ*, Studia Math.**88**(1988), no. 1, 69–84. MR**932007****[BKM]**J. Bourgain, B. Klartag, and V. Milman,*Symmetrization and isotropic constants of convex bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 101–115. MR**2087154**, 10.1007/978-3-540-44489-3_10**[Bo1]**J. Bourgain,*On the distribution of polynomials on high-dimensional convex sets*, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR**1122617**, 10.1007/BFb0089219**[Bo2]**J. Bourgain,*On the isotropy-constant problem for “PSI-2”-bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 114–121. MR**2083391**, 10.1007/978-3-540-36428-3_9**[BM]**J. Bourgain and V. D. Milman,*New volume ratio properties for convex symmetric bodies in 𝑅ⁿ*, Invent. Math.**88**(1987), no. 2, 319–340. MR**880954**, 10.1007/BF01388911**[D]**S. Dar,*Remarks on Bourgain’s problem on slicing of convex bodies*, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 61–66. MR**1353449****[Gi]**A.GIANNOPOULOS,*Notes on isotropic convex bodies*, http://itia.math.uch.gr/ apostolo/notes.html.**[GM1]**A. A. Giannopoulos and V. D. Milman,*On the diameter of proportional sections of a symmetric convex body*, Internat. Math. Res. Notices**1**(1997), 5–19. MR**1426731**, 10.1155/S1073792897000020**[GM2]**Apostolos A. Giannopoulos and Vitali D. Milman,*How small can the intersection of a few rotations of a symmetric convex body be?*, C. R. Acad. Sci. Paris Sér. I Math.**325**(1997), no. 4, 389–394 (English, with English and French summaries). MR**1467092**, 10.1016/S0764-4442(97)85622-4**[GM3]**A. A. Giannopoulos and V. D. Milman,*Mean width and diameter of proportional sections of a symmetric convex body*, J. Reine Angew. Math.**497**(1998), 113–139. MR**1617429**, 10.1515/crll.1998.036**[GMT]**Apostolos Giannopoulos, Vitali D. Milman, and Antonis Tsolomitis,*Asymptotic formulas for the diameter of sections of symmetric convex bodies*, J. Funct. Anal.**223**(2005), no. 1, 86–108. MR**2139881**, 10.1016/j.jfa.2004.10.006**[H]**Douglas Hensley,*Slicing convex bodies—bounds for slice area in terms of the body’s covariance*, Proc. Amer. Math. Soc.**79**(1980), no. 4, 619–625. MR**572315**, 10.1090/S0002-9939-1980-0572315-5**[KLS]**R. Kannan, L. Lovász, and M. Simonovits,*Isoperimetric problems for convex bodies and a localization lemma*, Discrete Comput. Geom.**13**(1995), no. 3-4, 541–559. MR**1318794**, 10.1007/BF02574061**[Kl1]**Bo’az Klartag,*A geometric inequality and a low 𝑀-estimate*, Proc. Amer. Math. Soc.**132**(2004), no. 9, 2619–2628 (electronic). MR**2054787**, 10.1090/S0002-9939-04-07484-2**[Kl2]**B. KLARTAG,*On convex perturbations with a bounded isotropic constant*, to appear in Geom. and Funct. Anal.**[LT]**A. E. Litvak and N. Tomczak-Jaegermann,*Random aspects of high-dimensional convex bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 169–190. MR**1796719**, 10.1007/BFb0107214**[MP]**V. D. Milman and A. Pajor,*Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed 𝑛-dimensional space*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR**1008717**, 10.1007/BFb0090049**[MS]**Vitali D. Milman and Gideon Schechtman,*Asymptotic theory of finite-dimensional normed spaces*, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR**856576****[Pa1]**G. Paouris,*On the isotropic constant of non-symmetric convex bodies*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 239–243. MR**1796722**, 10.1007/BFb0107217**[Pa2]**Grigoris Paouris,*Concentration of mass on isotropic convex bodies*, C. R. Math. Acad. Sci. Paris**342**(2006), no. 3, 179–182 (English, with English and French summaries). MR**2198189**, 10.1016/j.crma.2005.11.018**[Pa3]**G. PAOURIS,*Concentration of mass in convex bodies*, to appear in Geom. Funct. Anal.**[Pi]**Gilles Pisier,*The volume of convex bodies and Banach space geometry*, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR**1036275**

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Additional Information

**Jesús Bastero**

Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Email:
bastero@unizar.es

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08693-5

Keywords:
Asymptotic geometric analysis,
diameter of sections

Received by editor(s):
September 14, 2005

Received by editor(s) in revised form:
February 8, 2006, and February 14, 2006

Published electronically:
January 5, 2007

Additional Notes:
The author was partially supported by DGA (Spain), MCYT (Spain) MTM2004-03036 and EU Project MRTN-CT-2004-511953 and is grateful to the Erwin Schrödinger Institut in Wien, where part of this work was carried out

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.