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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Functions with isotropic sections


Authors: Ioannis Purnaras and Christos Saroglou
Journal: Trans. Amer. Math. Soc. 374 (2021), 3007-3024
MSC (2020): Primary 52A40
DOI: https://doi.org/10.1090/tran/8321
Published electronically: February 8, 2021
MathSciNet review: 4223041
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Abstract: We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if $n\geq 3$, $g:\mathbb {S}^{n-1}\to \mathbb {R}$ is a bounded measurable function, $U$ is an open connected subset of $\mathbb {S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then $\mathcal {C}(g)|_U=c+\langle a,\cdot \rangle$ and $\mathcal {R}(g)|_U=c’$, for some fixed constants $c,c’\in \mathbb {R}$ and for some fixed vector $a\in \mathbb {R}^n$. Here, $\mathcal {C}(g)$ denotes the cosine transform and $\mathcal {R}(g)$ denotes the Funk transform of $g$. However, we show that an even $g$ does not need to be equal to a constant almost everywhere in $U^\perp \coloneq \bigcup _{u\in U}(\mathbb {S}^{n-1}\cap u^\perp )$. For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.


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Additional Information

Ioannis Purnaras
Affiliation: Department of Mathematics, University of Ioannina, Ioannina, 45110 Greece
MR Author ID: 311798
Email: ipurnara@uoi.gr

Christos Saroglou
Affiliation: Department of Mathematics, University of Ioannina, Ioannina, 45110 Greece
MR Author ID: 915316
ORCID: 0000-0001-5471-5560
Email: csaroglou@uoi.gr, christos.saroglou@gmail.com

Received by editor(s): June 21, 2020
Received by editor(s) in revised form: September 10, 2020
Published electronically: February 8, 2021
Article copyright: © Copyright 2021 American Mathematical Society