4 problems

Let $m>3$, let $a_1,\dots,a_m>0$ satisfy $\sum_{i=1}^m a_i=1$, and let $w\in\mathbb{R}^{n+1}$ be the vector defined in the paper. Set $\overline{w}^{(i)}=w/a_i$. Consider measurabl…

Propeller partition conjecture. For every $k\ge 3$, an optimal partition is

Sharpness conjecture. The displayed inequality is sharp for every $k\ge 4$.

Sharp-threshold conjecture. Assuming UGC, $\frac{8\pi}{9}\left(1-\frac{1}{k}\right)$ is the sharp approximability threshold for $\mathrm{\bf Clust}(A|I_k)$ for every $k\ge 3$.