7 problems

Connectedness conjecture. Isoperimetric clusters have simply connected chambers.

Peanut and tailor's chalk conjectures. Among $(2,2)$-clusters with area vector $(1,1,\infty,\infty)$, the minimizer is the peanut shape. Among $(1,3)$-clusters with area vector…

Standard lens cluster conjecture. The minimizer is the standard lens cluster: a finite-area chamber bounded by circular arcs attached to a single axis, with two triple junctions at…

Let $\mathcal C(1,1,1,1)$ denote the stationary planar clusters with four regions of area $1$, and let a sandwich-topology cluster be one with the topology described in the source.…

Let $\boldsymbol E=(E_1,E_2,E_3,E_4)$ be a perimeter-minimizing planar cluster with four equal-area regions, and suppose it has the sandwich topology. Symmetry conjecture. The mini…

Let $N$ be a positive integer, let $\boldsymbol a\in\mathbb{R}^N_+$ be a vector of prescribed positive areas, and let $\mathcal M(\boldsymbol a)$ denote the set of perimeter-minimi…

Let a perimeter-minimizing planar $N$-cluster have chambers of equal volume, and suitably normalize the cluster so that its total area is fixed. Morgan–Heppes conjecture. As $N$ te…