On convex bodies with constant non-central sections

TL;DR

This paper proves that symmetric convex bodies of revolution in four dimensions with specific constant-area tangent hyperplane sections are Euclidean balls, under certain arithmetic conditions related to their section areas.

Contribution

It establishes a uniqueness result for convex bodies with constant non-central sections in four dimensions, linking geometric properties to number-theoretic conditions.

Findings

Such bodies are Euclidean balls under the given conditions.

The set of section areas satisfying the conditions has positive Hausdorff dimension.

The result connects geometric properties with arithmetic and continued fraction conditions.

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 $\frac 1{\pi} \arctan((\frac{3A}{4\pi})^{1/3})$ 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.