5 problems

Langerman's conjecture. For any $nd$ masses on $\mathbb{R}^d$, there exist $n$ hyperplanes whose induced chessboard partition is balanced for every one of the $nd$ masses.

Soberón–Takahashi conjecture. Any $d+k-1$ finite absolutely continuous measures on $\mathbb{R}^d$ can be simultaneously bisected by the chessboard coloring induced by $k$ or fewer…

Optimality conjecture. The result referred to as the fairy bread sandwich should be optimal for every such $k$, $d$, and permutation $\pi$.

Let a mass assignment in $\mathbb{R}^d$ assign continuously to each relevant linear subspace a mass distribution on that subspace, and let a $(k-1)$-horizontal linear subspace mean…

Let a mass distribution in $\mathbb{R}^d$ be a measure for which the bisection notion is defined. A set of $n$ hyperplanes bisects a mass distribution when, after orienting the hyp…