7 problems

There exist an absolutely continuous probability measure $\mu$ on $\mathbb{R}^2$ and a number $t\in(0,1/4)$ such that, for every pair of perpendicular lines in the plane, the four…

Ramos's conjecture. The triple $(d,k,m)$ is a solution if and only if

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 $\pi$.

Kano's conjecture. There exists a curve in $\mathds{R}^2$ formed by horizontal and vertical segments, with at most $m-1$ turns, that bisects every measure.

Langerman's conjecture. For every such family of measures, there exists a set of $n$ hyperplanes whose induced chessboard coloring splits every measure into two equal parts.

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…