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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

Information

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

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