## Functions with isotropic sections

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- by Ioannis Purnaras and Christos Saroglou PDF
- Trans. Amer. Math. Soc.
**374**(2021), 3007-3024 Request permission

## 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.## References

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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
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3007-3024 - MSC (2020): Primary 52A40
- DOI: https://doi.org/10.1090/tran/8321
- MathSciNet review: 4223041