Functions with isotropic sections

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.