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Barker–Larman problem

For each n≥2n\ge 2, let Bn⊆Rn\mathbb B_n\subseteq\mathbb R^n be the Euclidean unit ball and let K⊆RnK\subseteq\mathbb R^n be a convex body with Bn⊆K\mathbb B_n\subseteq K. If vol⁡n−1(K∩H)\operatorname{vol}_{n-1}(K\cap H) is independent of the affine hyperplane HH tangent to Bn\mathbb B_n, must KK be a Euclidean ball?

References

Primary source

arXiv

Additional references

convex-geometrygeometric-analysis
Posted · edited

Progress summary

Refreshed
Claimed progress

A new study produces extraordinarily close approximate counterexamples in four dimensions, but an actual counterexample is not yet established.

The Barker–Larman problem asks whether convex bodies with matching sections along every hyperplane tangent to a containing unit ball must be identical. The general question remains open, including in dimension four.

Known results

  • Santaló proved the planar case in 1951.
  • Barker and Larman proved a local planar result and an analogous result in odd dimensions.
  • The assertion is affirmative for sections by subspaces of codimension greater than 11.
  • A restricted four-dimensional body-of-revolution result proves uniqueness for constants in an explicitly constructed infinite set.

September 2026 approximate counterexamples

A September 2026 preprint constructs arbitrarily high-order approximate counterexamples in dimension four and gives an explicit example with section oscillation below 3×10−73\times 10^{-7}. It does not establish a genuine counterexample: exact convergence of the required infinite series remains open.

Current status (as of September 2026): The general problem remains open; four-dimensional approximate counterexamples are claimed, but exact convergence and any genuine counterexample are unproved.

Sources

Solutions 0

No solutions have been posted yet.