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 an even bounded measurable function, $U$ is an open subset of $\mathbb{S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)|_U=c'$, for some fixed constants $c,c'\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$. Here, ${\cal C}(g)$ denotes the cosine transform and ${\cal R}(g)$ denotes the Funk transform of $g$. However, we show that $g$ does not need to be equal to a constant almost everywhere in $U^\perp:=\bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp)$. 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.