16 problems

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubbl…

Let $A_0=3\sqrt{3}/2$, and let an infinite planar cluster consist of equal-area regions of area $A_0$. Let $E$ be the area of its complement, with $0\leq E\leq\infty$. A cluster is…

For a cluster enclosing $n$ regions of prescribed areas $a_1,\ldots,a_n$, call it minimizing if it has least perimeter among clusters enclosing and separating regions with those ar…

Let $p$ denote the least perimeter of a planar cluster enclosing and separating $n$ equal-area regions, with each region having area $A_0=3\sqrt{3}/2$. Asymptotic circularity conje…

A planar cluster encloses and separates $n$ regions of prescribed areas, with boundaries consisting of circular arcs meeting three at a time at $120$ degrees. Connectivity conjectu…

Let $\mathbb{G}^n$ denote Gaussian space, and let $\Omega^{k+1}[\mathbb{G}^n]$ be a standard simplicial $k$-bubble, meaning a $(k+1)$-partition obtained by translating a centered s…

Let $\mathbb{M}^n$ be one of $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For $2 \leq k \leq n+1$, let a standard $k$-bubble be the standard partition of $\mathbb{M}^n$ into…

Let $k$le$d+1$ and consider the problem of separating $k$ prescribed volumes in $d$-dimensional Euclidean space with least interface area. A standard $k$-bubble is the conjectured…

Consider the improper planar partitioning problem with one finite-measure chamber of prescribed area and three infinite-measure chambers, with locally perimeter minimizing understo…

Let $q\geq 2$ finite-volume chambers be partitioned in $\mathbb{R}^n$, with prescribed volumes, together with the surrounding chamber, and let perimeter be the total interface area…

Morgan–Sullivan's standard $k$-bubble conjecture. The standard $k$-bubble in $\mathbb{R}^{n}$ is the unique minimizer enclosing $k$ regions of prescribed volume.

Spherical minimizer conjecture. The area-minimizing surface enclosing volume $V$ is a sphere.

Bubbletoy minimality conjecture. The two pictured bubbletoys are the minimal planar configurations for four and five equal-area bubbles, respectively.

Let $B$ be a perimeter-minimizing $n$-bubble in $\mathbb{R}$ with continuous density that is nonincreasing on $(-\infty,0]$ and nondecreasing on $[0,\infty)$. If one bubble has uni…

Let $N$ and $n$ be positive integers. Consider a perimeter-minimizing $n$-bubble in $\mathbb{R}^N$ with a smooth, radially symmetric, strictly log-convex density. A standard nested…

Let $n$ be a positive integer, and let $k_1,\dots,k_m$ be positive integers satisfying … A standard nested bubble of type $(k_1,\dots,k_m)$ is a perimeter-minimizing $n$-bubble for…