An approximate counterexample to the Barker--Larman problem in dimension $4$

Abstract

The Barker--Larman problem asks if a convex body $K\subseteq {\mathbb{R}}^n$ containing the Euclidean ball ${\mathbb{B}}_n$, such that all the sections of $K$ by hyperplanes tangent to ${\mathbb{B}}_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking ${\lambda }_0=4\sqrt{3}\pi$ and any $N\in \mathbb{N}$, we obtain the existence of a family of convex bodies $K_{\lambda ,N}$ with $\lambda \in ({\lambda }_0-r_N,{\lambda }_0+r_N)$, such that the sections of $K_{\lambda ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c|\lambda -{\lambda }_0{|}^{\frac{N+1}{2}}$ of $\lambda$, while the difference between outradius and inradius of $K_{\lambda ,N}$ is larger than $C|\lambda -{\lambda }_0|$. The bodies $K_{\lambda ,N}$ are constructed via radial functions as \[ρ_{K_{λ,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(λ-λ_0)^n}{n!} φ_n(t) \right)^{-1},\] where $t\in [0,2\pi ),\lambda \in \mathbb{R}$ and ${\phi }_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N\to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3\times 10^{-7}$.