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 \[\rho_{K_{\lambda,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(\lambda-\lambda_0)^n}{n!} \varphi_n(t) \right)^{-1},\] where $t \in [0,2\pi), \lambda \in \mathbb R$ and $\varphi_n$ 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 $3 \times 10^{-7}$.