For every integer n≥ 1 and every finite, nonzero Borel measure mu on R^n that is absolutely continuous with respect to Lebesgue measure, there exist n affine hyperplanes H_1,…,H_n such that each of the 2^n regions determined by them has measure mu(R^n)/2^n. Equivalently, for every sign vector varepsilon∈+1,-1^n, the corresponding region bigcap_i=1^n H_i^varepsilon_i has measure mu(R^n)/2^n.

Grünbaum’s 1960 hyperplane equipartition conjecture; Ramos’ hyperplane equipartition conjecture

Grünbaum’s 1960 conjecture asks whether every continuous mass in $\mathbb{R}^n$ can be divided into $2^n$ equal parts by $n$ hyperplanes. Ramos proposed a broader formula for the minimum number of hyperplanes needed to equipartition several masses.

Known results

$n \le 3$ : affirmative for Grünbaum’s problem (Hadwiger, 1966).
$n \ge 5$ : negative for Grünbaum’s problem (Avis, 1984; Ramos, 1996).
$n=4$ : previously recorded as open, with $4 \le \Delta(1,4) \le 5$ .
Ramos’ general formula was established only in special cases.

August 24, 2026 claimed counterexample

A linked arXiv manuscript claims a counterexample in $\mathbb{R}^4$ , thereby settling the last unresolved case of Grünbaum’s conjecture negatively and disproving Ramos’ broader conjecture. The argument uses computer-assisted subdivision and Bernstein-coefficient verification, but remains unrefereed.

Current status (as of August 2026): The classical cases $n \le 3$ and $n \ge 5$ are settled, while the new negative $n=4$ result and the claimed disproof of Ramos’ conjecture are unverified.

Grünbaum’s 1960 hyperplane equipartition conjecture; Ramos’ hyperplane equipartition conjecture

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