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

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 $c\mathrm{\mid }\lambda -{\lambda }_0{\mathrm{\mid }}^{\frac{N+1}{2}}$ of $\lambda$, while the difference between outradius and inradius of $K_{\lambda ,N}$ is larger than $C\mathrm{\mid }\lambda -{\lambda }_0\mathrm{\mid }$. The bodies $K_{\lambda ,N}$ are constructed via radial functions as $${\rho }_{K_{\lambda ,N}}(t)=\cos {(\sum _{n=0}^N\frac{(\lambda -{\lambda }_0{)}^n}{n!}{\phi }_n(t))}^{-1},$$ 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 $N\to \mathrm{\infty }$ (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 $3\times {10}^{-7}$.