7 problems

Let $k\geq 2$. Choose $k$ points independently according to the volume measure of a compact Brownian surface of fixed genus, and consider the associated $k$ Voronoï cells. Their vo…

Let $V_{\alpha}(\bar x)$ be the potential of a generic configuration of point charges of the same sign, and let $a_{\alpha}^j$ denote the number of its critical points of index $j$…

Let a finite configuration of positive charges be given, and let $\alpha>0$. Consider its nondegenerate equilibrium points and its effective Voronoi cells. Effective Voronoi-cell b…

Gabrielov–Novikov–Shapiro conjecture. For $n$ unit point charges in generic position, and all $p\geq 1$, the number of index-$j$ equilibria of $V_p$ is at most $\#_j$.

Voronoi degree conjecture. The Voronoi degree of a generic hypersurface of degree $d$ in $\mathbb{R}^n$ equals

Let $P$ be a packing of unit spheres in ${\mathbb R}^4$, and consider its Voronoi decomposition. Let a Voronoi cell be the region associated with one sphere of the packing, and let…

Let the unit disc be equipped with the uniform probability distribution. For an optimal set of $n$-means, consider the associated Voronoi regions. Conjecture on optimal quantizers…