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Petty projection inequality on the sphere and on the hyperbolic space

Part of: Classical differential geometry Classical measure theory General convexity

Published online by Cambridge University Press:  28 August 2025

Youjiang Lin  and
Yuchi Wu
Youjiang Lin
Affiliation:
School of Mathematical Sciences, Hebei Normal University , Shijiazhuang 050024, China Dipartimento di Matematica e Informatica “U. Dini”, Universitá di Firenze , Firenze 50121, Italy e-mail: youjiang.lin@unifi.it; yjlin@hebtu.edu.cn
Yuchi Wu*
Affiliation:
School of Mathematical Sciences, Key Laboratory of MEA(Ministry of Education), and Shanghai Key Laboratory of PMMP, East China Normal University , Shanghai 200241, China
*
e-mail: ycwu@math.ecnu.edu.cn
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Abstract

We define a spherical and hyperbolic analog to the Euclidean projection body for star bodies via the gnomonic projection from the unit sphere and stereographic projection in the hyperbolid model of hyperbolic space. We then prove a spherical and hyperbolic projection inequality for these notions by using an adaption of Steiner symmetrization for spherical, respectively, hyperbolic, star bodies.

Keywords

Projection bodyPetty projection inequalityspherical and hyperbolic spacesSteiner symmetrizationisoperimetric inequality

MSC classification

Primary: 52A55: Spherical and hyperbolic convexity
Secondary: 28A75: Length, area, volume, other geometric measure theory 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53A35: Non-Euclidean differential geometry

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 28
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Research of the first author is supported by NSFC 11971080, NSFC 12371137, Hebei Normal University Doctoral Research Start-up Fund L2025B50 and Jiangxi Provincial Natural Science Foundation 20232BAB201005. Research of the second author (corresponding author) is supported by Science and Technology Commission of Shanghai Municipality 22DZ2229014 and Youth Fund of the National Natural Science Foundation of China NSFC 12401067.

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