An approximate counterexample to the Barker--Larman problem in dimension $4$
Abstract
The Barker--Larman problem asks if a convex body K ⊆ ℝn containing the Euclidean ball 𝔹n, such that all the sections of K by hyperplanes tangent to 𝔹n 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 λ0 = 4 √3π and any N ∈ ℕ, we obtain the existence of a family of convex bodies Kλ,N with λ∈ (λ0−rN, λ0 + rN), such that the sections of Kλ,N by hyperplanes tangent to the Euclidean ball, have area within c |λ− λ0|N+1 of λ, while the difference between outradius and inradius of Kλ,N is larger than C |λ− λ0|. The bodies Kλ,N are constructed via radial functions as ρKλ,N(t) = cos( ∑n=0N ((λ−λ0)n)/n! φn(t) )−1, where t ∈ [0,2π), λ∈ ℝ and φn are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when N → ∞ (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 × 10−7.
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BibTeX
@misc{haddad2026approximate,
title = {{An approximate counterexample to the Barker--Larman problem in dimension $4$}},
author = {J. Haddad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10257},
eprint = {2609.10257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10257},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An approximate counterexample to the Barker--Larman problem in dimension $4$
AU - J. Haddad
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.10257
UR - https://arxiv.org/abs/2609.10257
AB - The Barker--Larman problem asks if a convex body K ⊆ ℝ^n containing the Euclidean ball 𝔹_n, such that all the sections of K by hyperplanes tangent to 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 λ_0 = 4 √3π and any N ∈ ℕ, we obtain the existence of a family of convex bodies K_λ,N with λ∈ (λ_0−r_N, λ_0 + r_N), such that the sections of K_λ,N by hyperplanes tangent to the Euclidean ball, have area within c |λ− λ_0|^N+1 of λ, while the difference between outradius and inradius of K_λ,N is larger than C |λ− λ_0|. The bodies K_λ,N are constructed via radial functions as ρ_K_λ,N(t) = cos( ∑_n=0^N ((λ−λ_0)^n)/n! φ_n(t) )^−1, where t ∈ [0,2π), λ∈ ℝ and φ_n are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when N → ∞ (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 × 10^−7.
ER -