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ON THE INTEGRAL FORMULAS OF CROFTON AND HURWITZ RELATIVE TO THE VISUAL ANGLE OF A CONVEX SET

Part of: Classical differential geometry General convexity

Published online by Cambridge University Press:  21 May 2019

Julià Cufí ,
Eduardo Gallego  and
Agustí Reventós
Julià Cufí
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain email jcufi@mat.uab.cat
Eduardo Gallego
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain email egallego@mat.uab.cat
Agustí Reventós
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain email agusti@mat.uab.cat
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Abstract

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.

MSC classification

Primary: 52A10: Convex sets in $2$ dimensions (including convex curves) 53A04: Curves in Euclidean space
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 874 - 896
Copyright
Copyright © University College London 2019 

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Footnotes

The authors were partially supported by grants 2017SGR358, 2017SGR1725 (Generalitat de Catalunya) and MTM2015-66165-P (Ministerio de Economía y Competitividad).

References

Crofton, M. W., On the theory of local probability. Philos. Trans. R. Soc. Lond. A 158 1868, 181–199.Google Scholar
Cufí, J. and Reventós, A., A lower bound for the isoperimetric deficit. Elem. Math. 71(4) 2016, 156–167.10.4171/EM/312Google Scholar
Gauss, C. F., Disquisitiones generales circa series infinitam. In Werke (Cambridge Library Collection – Mathematics), Cambridge University Press (Cambridge, 2011), 123–162. First published in 1866.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, Academic Press (New York–London–Toronto, 1980).Google Scholar
Groemer, H., Geometric applications of Fourier series and spherical harmonics. In Encyclopedia of Mathematics and its Applications, Vol. 61, Cambridge University Press (Cambridge, 1996).Google Scholar
Hurwitz, A., Sur quelques applications geometriques des séries de Fourier. Ann. Sci. Éc. Norm. Supér. 19 1902, 357–408.10.24033/asens.514Google Scholar
Masotti, G., La Geometria Integrale. Rend. Sem. Mat. Fis. Milano 25 1955, 164–231.Google Scholar
Masotti, G., Sulla Geometria Integrale: Generalizzazione Di Formiule Di Crofton, Lebesgue E Santalo. Rev. Un. Mat. Argentina 17 1955, 125–134.Google Scholar
Santaló, L., Integral geometry and geometric probability, 2nd edn., Cambridge University Press (Cambridge, 2004).10.1017/CBO9780511617331Google Scholar
Schneider, R., Convex bodies: the Brunn–Minkowski theory (Encyclopedia of Mathematics and its Applications 151 ), Cambridge University Press (Cambridge, 2014).Google Scholar