Barker–Larman Conjecture: Central symmetry of sections tangent to an inscribed sphere

Prove that for a convex body K ⊂ ℝ^n (n ≥ 3), if there exists a sphere B contained in the interior of K such that for every supporting hyperplane Π of B the section Π ∩ K is centrally symmetric, then K is an ellipsoid.

Background

Another classical open problem cited by the authors is due to Barker and Larman. It concerns whether central symmetry of all sections of a convex body by hyperplanes supporting an inscribed sphere implies that the body is an ellipsoid.

The paper’s results on ellipsoidal sections tangent to a sphere are presented as steps toward addressing this conjecture.

References

... and, on the other hand, the J. A. Barker and D. G. Larman's Conjecture: Let $K \subset R$ be a convex body, $n \geq 3$. If there exists a sphere $B\subset int K$ such that, for every supporting hyperplane $\Pi$ of $B$, the section $\Pi \cap K$ is centrally symmetric, then $K$ is an ellipsoid.

— On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body  (2401.03983 - Morales-Amaya et al., 2024) in Introduction

Unfortunately, we were not able to show that the power series is convergent. Instead we will study the partial sums, as approximate counterexamples for the Barker--Larman problem.

— An approximate counterexample to the Barker--Larman problem in dimension $4$  (2609.10257 - Haddad, 9 Sep 2026) in Section 6, “Convergence of the Power Series” (Concluding Remarks)

We conjecture here that these counterexamples exist around \lambda_q>0 where q = \frac 1\pi \arctan((\lambda_q/\omega_3){1/3}) \in \mathbb Q can be written with an odd denominator.

— An approximate counterexample to the Barker--Larman problem in dimension $4$  (2609.10257 - Haddad, 9 Sep 2026) in Section 6, “Choice of the Bifurcation point” (Concluding Remarks)