On convex bodies with constant non-central sections

Abstract

We prove that if $C$ is a symmetric convex body of revolution in ${\mathbb{R}}^4$ containing the unit Euclidean ball ${\mathbb{B}}_4$, such that the sections of $C$ by hyperplanes tangent to ${\mathbb{B}}_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac{1}{\ddot{I}}€\arctan ((\frac{3A}{4\ddot{I}€}{)}^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.