3 problems

Let $\rho\in(0,1)$ and $n\geq k!+k-1$. Consider partitions $\Omega_1,\ldots,\Omega_k$ of $\mathbb{R}^n$ with Gaussian measure $\gamma_n(\Omega_i)=1/k$ that maximize the ranked-choi…

Gaussian partition conjecture. For every integer $k\ge 2$,

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