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 $n\ge 3$, $g:{\mathbb{S}}^{n-1}\to \mathbb{R}$ 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 $\mathcal{C}(g){|}_U=c+⟨a,\cdot ⟩$ and $\mathcal{R}(g){|}_U=c^{\prime }$, for some fixed constants $c,c^{\prime }\in \mathbb{R}$ 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 $g$ 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.

The previous version contained an error in the statement of Theorem 1.2. Necessary changes have been made. 19 pages