23 problems

Consider the Kelvin problem in dimension $d=2$: partition the Euclidean plane into cells of equal area while minimizing the total interface length. The Honeycomb tiling is the regu…

Global Holmes–Thompson isoperimetric conjecture. The circle $c_0$ encloses the maximal Holmes–Thompson area among all such curves.

Let $f=e^h$ be a smooth, radial, log-convex density on $\mathbb{R}^n$, and define the weighted volume and perimeter of a set $A\subset\mathbb{R}^n$ by … A set is an isoperimetric r…

Spherical isoperimetric-surface conjecture. Isoperimetric surfaces in ${\mathrm{Nil}\,}$ are homeomorphic to spheres.

Isoperimetric-sphere conjecture. The constant-mean-curvature spheres of revolution in ${\mathrm{Nil}\,}$ are isoperimetric surfaces.

Let a space-filling polyhedron be a convex polyhedron that tiles three-dimensional space by translations, and restrict to such polyhedra of unit volume. The truncated octahedron is…

Let $M$ and $N$ be compact Riemannian manifolds. For a fixed volume fraction, consider the product of $N$ with a sufficiently small homothetic copy of $M$, equipped with the corres…

Mask conjecture. A minimizer is a "mask": its boundary is composed of arcs of circles with three different curvatures.

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…

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 $M$ be an $(n+1)$-dimensional Riemannian manifold endowed with a codimension-one complete foliation $\mathcal{F}$. Assume that each leaf of $\mathcal{F}$ is a closed, totally u…

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

Let $H^1$ be the Riemannian Heisenberg group. For $R>0$, let $E_R\subset H^1$ be the region bounded by the centered constant-mean-curvature sphere $\Sigma_R$, and let…

Let $H^1$ be the Riemannian Heisenberg group, and let $\Sigma_R\subset H^1$, with $R>0$, be the family of centered constant-mean-curvature spheres foliating $H^1\setminus\{0\}$. Le…

Spherical-boundary conjecture. In the model spaces $\mathbb{M}(\kappa)$, every isoperimetric region is bounded by a spherical surface $\mathcal{S}_\lambda(p)$.

Radial increasing density conjecture. Isoperimetric sets exist for all volumes.

Let the $3$-sphere be endowed with a metric of positive Ricci curvature. An isoperimetric region is a region minimizing area among regions enclosing a prescribed volume. Ros's conj…

Round-sphere stability conjecture. In $\mathbf{R}^n$, round $m$-spheres $S_0$ are the only smooth stable surfaces $S$ for prescribed volume … or prescribed $\Omega$-volume for cons…

Let $f$ be a radial, increasing density on $\mathbb{R}^n$. An isoperimetric set is a set minimizing weighted perimeter among sets of prescribed weighted volume. Existence conjectur…

Let $\mathbb{H}^n$ be the Heisenberg group, and let $\mathbb{S}_{\mathbb{H}^n}$ denote its unit Isoperimetric Profile. For a set with finite $\mathit{H}$-perimeter, consider the…

Consider the isoperimetric problem in $\operatorname{Nil}_3$ and in the universal cover of $\operatorname{PSL}(2,\mathbb{R})$. The canonical rotational spheres are the rotational C…

Let $\mathrm{Sol}_3$ be the three-dimensional Sol geometry, and let $H>0$. A constant mean curvature (CMC) $H$ sphere is an immersed sphere in $\mathrm{Sol}_3$ with constant mean c…