Bárány’s conjecture on uneven orthogonal mass partitions
There exist an absolutely continuous probability measure on and a number such that, for every pair of perpendicular lines in the plane, the four regions that they determine cannot be cyclically ordered as with and .
References
Primary source
Additional references
- Counterexamples and symmetry for uneven orthogonal mass partitions in the plane — arXiv — Leonardo Martínez-Sandoval
Progress summary
A September 2026 preprint claims a robust counterexample, including a finite example, disproving Bárány’s conjecture, but the result has not been independently verified.
Bárány’s conjecture asks whether an absolutely continuous planar mass can always be cut by two perpendicular lines into prescribed regions of masses ; the measure version was publicly posed by Bárány in connection with a question of Grünbaum.
Known results
For convex bodies, the analogous assertion is conjectured positive; a positive result is known for sufficiently elongated bodies, measured by the ratio of diameter to minimum width. The survey records no previous resolution of the absolutely continuous-measure conjecture.
September 2026 claimed counterexample
Leonardo Martínez-Sandoval’s preprint claims the negative answer in a robust class of planar measures, with failure persisting under regularity, positivity, symmetry, and proximity to the standard Gaussian. It also claims a finite -point counterexample for the pattern . The preprint’s claims are not independently verified in the retrieved sources.
Current status (as of September 2026): Bárány’s conjecture is claimed disproved by the new preprint, but that counterexample and its finite realization remain unverified; related convex-body formulations remain distinct and open.
Solutions 0
No solutions have been posted yet.