A Support Characterization for Functions on the Unit Sphere with Vanishing Integrals Arising from Tangent Planes to a Given Surface

TL;DR

This paper characterizes the support of functions on the unit sphere based on the vanishing of their integrals over specific subspheres related to tangent hyperplanes of a convex surface.

Contribution

It provides a new support characterization for functions on the sphere using integrals over tangent and perturbed tangent hyperplanes of a convex surface.

Findings

Support of the function is determined by vanishing integrals over tangent subspheres.

Supports are characterized under conditions of local convexity and regularity.

The method extends previous integral geometry results to more general surfaces.

Abstract

Let $\Sigma$ be an axially symmetric, smooth, closed hypersurface in $\Bbb R^{n + 1}$ with a simply connected interior which is contained inside the unit sphere $\Bbb S^{n}$. For a continuous function $f$, which is defined on $\Bbb S^{n}$, the main goal of this paper is to characterize the support of $f$ in case where its integrals vanish on subspheres obtained by intersecting $\Bbb S^{n}$ with the tangent hyperplanes of a certain subdomain $\mathcal{U}\subset\Sigma$ of $\Sigma$. We show that the support of $f$ can be characterized in case where its integrals also vanish on subspheres obtained by intersecting $\Bbb S^{n}$ with hyperplanes obtained by infinitesimal perturbations of the tangent hyperplanes of $\mathcal{U}$ and where $\mathcal{U}$ satisfies some regularity condition which implies local convexity.