4 problems

Asymptotic discrepancy-ratio conjecture. Based on numerical evidence, it was conjectured that

Optimality conjecture. The partition considered in the paper is best possible among all partitions generated from parallel lines.

For each $N$, let $\mathcal{P}_N$ be a point set of $N$ random points and let $\mathcal{P}_{\mathbf{\Omega}_{\ast}^{(N)}}$ be the point set obtained from the equivolume partition…

Let $\mathbf{v}=(v_1,v_2,v_3)$ satisfy $v_1<v_2<v_3$ with each $v_i\in[0,\sqrt{2}]$. Let $\mathbf{\Omega}_{\mathbf{v}}^{(4)}$ be the associated four-cell partition, let…