publication . Article . Preprint . 2015

Distribution functions of sections and projections of convex bodies

Jaegil Kim; Vladyslav Yaskin; Artem Zvavitch;
Open Access
  • Published: 22 Dec 2015 Journal: Journal of the London Mathematical Society, volume 95, pages 52-72 (issn: 0024-6107, Copyright policy)
  • Publisher: Wiley
Abstract
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get the information about the sizes of sections (or projections), and not about the corresponding directions. In this paper we study to what extent the distribution function of the areas of central sections (or projections) of a convex body can be used to derive some information about the body, its volume, etc.
Persistent Identifiers
Subjects
free text keywords: Mathematics - Metric Geometry, Pure mathematics, Distribution function, Mathematics, Regular polygon
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
,
NSF| Harmonic Analysis in Convex Geometry
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1101636
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
27 references, page 1 of 2

[A] D. Alonso-Guti´errez, On a reverse Petty projection inequality for projections of convex bodies, Adv. Geom. 14 (2014), 215-223.

[Ba] K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351-359.

[Ba1] K. Ball, Volumes of sections of cubes and related problems, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, Springer, Berlin, 1989, 251-260.

[Bo1] J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), 1467-1476. [OpenAIRE]

[Bo2] J. Bourgain, Geometry of Banach spaces and harmonic analysis, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 871-878. [OpenAIRE]

[BM] J. Bourgain, V. D, Milman, New volume ratio properties for convex symmetric bodies in Rn, Invent. Math. 88 (1987), no. 2, 319-340. [OpenAIRE]

[BGVV] S. Brazitikos, A. Giannopoulos, P. Valettas and B. Vritsiou, Geometry of isotropic log-concave measures, Amer. Math. Soc., Providence RI, 2014. [OpenAIRE]

[F] W. Feller, An Introduction to Probability Theory and its Applications, vol. 2. Second edition. John Wiley & Sons, 1971.

[Ga] R.J. Gardner, Geometric Tomography. Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 2006.

[Gr] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, New York, 1996.

[GS] I. M. Gelfand and G. E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York and London, 1964.

[GV] I.M. Gelfand and N. Ya. Vilenkin, Generalized functions, vol. 4. Applications of harmonic analysis, Academic Press, New York, 1964.

[GP] A. Giannopoulos, M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), 1-13.

[GYY] P. Goodey, V. Yaskin, and M. Yaskina, Fourier transforms and the Funk-Hecke theorem in convex geometry, J. London Math. Soc. (2) 80 (2009), 388-404.

[K] A. Koldobsky, Fourier Analysis in Convex Geometry, Math. Surveys and Monographs, AMS, Providence RI 2005.

27 references, page 1 of 2
Abstract
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get the information about the sizes of sections (or projections), and not about the corresponding directions. In this paper we study to what extent the distribution function of the areas of central sections (or projections) of a convex body can be used to derive some information about the body, its volume, etc.
Persistent Identifiers
Subjects
free text keywords: Mathematics - Metric Geometry, Pure mathematics, Distribution function, Mathematics, Regular polygon
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
,
NSF| Harmonic Analysis in Convex Geometry
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1101636
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
27 references, page 1 of 2

[A] D. Alonso-Guti´errez, On a reverse Petty projection inequality for projections of convex bodies, Adv. Geom. 14 (2014), 215-223.

[Ba] K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351-359.

[Ba1] K. Ball, Volumes of sections of cubes and related problems, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, Springer, Berlin, 1989, 251-260.

[Bo1] J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), 1467-1476. [OpenAIRE]

[Bo2] J. Bourgain, Geometry of Banach spaces and harmonic analysis, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 871-878. [OpenAIRE]

[BM] J. Bourgain, V. D, Milman, New volume ratio properties for convex symmetric bodies in Rn, Invent. Math. 88 (1987), no. 2, 319-340. [OpenAIRE]

[BGVV] S. Brazitikos, A. Giannopoulos, P. Valettas and B. Vritsiou, Geometry of isotropic log-concave measures, Amer. Math. Soc., Providence RI, 2014. [OpenAIRE]

[F] W. Feller, An Introduction to Probability Theory and its Applications, vol. 2. Second edition. John Wiley & Sons, 1971.

[Ga] R.J. Gardner, Geometric Tomography. Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 2006.

[Gr] H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, New York, 1996.

[GS] I. M. Gelfand and G. E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York and London, 1964.

[GV] I.M. Gelfand and N. Ya. Vilenkin, Generalized functions, vol. 4. Applications of harmonic analysis, Academic Press, New York, 1964.

[GP] A. Giannopoulos, M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), 1-13.

[GYY] P. Goodey, V. Yaskin, and M. Yaskina, Fourier transforms and the Funk-Hecke theorem in convex geometry, J. London Math. Soc. (2) 80 (2009), 388-404.

[K] A. Koldobsky, Fourier Analysis in Convex Geometry, Math. Surveys and Monographs, AMS, Providence RI 2005.

27 references, page 1 of 2
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