29 problems

Let $A_1$, $A_2$, and $A_3$ be positive real numbers. Consider planar graphs enclosing and separating finite areas $B_1$, $B_2$, and $B_3$ satisfying … A standard triple bubble is…

Let $A_0\geq A_2\geq A_1$ be areas, and let a flat torus have area $A_0+A_1+A_2$. A non-tiling minimizing double bubble is a perimeter-minimizing double bubble enclosing areas…

Let a minimizing graph be a least-perimeter partition of the unit disk into regions of prescribed areas. Connectedness conjecture. A minimizing graph separates the disk into connec…

Small-volume triple-bubble conjecture. The minimizers will look like the double bubbles of Figure 2 with a small ball attached, and the phase diagram will look just like Figure 3.

Complete-classification conjecture. The double bubbles of Figure 1 together with the Hexagonal Honeycomb of Figure 4 comprise the complete set of area-minimizing double bubbles for…

Very-long-torus Double Slab conjecture. In the special case of a very long $T^3$, the Double Slab is optimal for most volumes.

Concavity conjecture. The least area to enclose and separate two given volumes in the three-torus is a concave function of the volumes.

Ritoré–Ros conjecture. The optima for the isoperimetric problem in a cubic $T^3$ are the sphere, cylinder, and slab.

Let $N$ be the number of points and let $(P_m)$ denote the property that the maximizer represented by regular polygons has the specified geometric extremal behavior for $m$ points.…

Hauswirth–Pérez–Romon–Ros conjecture. The only solutions of the isoperimetric problem in $\mathbb{T}^2\times\mathbb{R}$ are spheres, cylinders, and pairs of horizontal planes.

Maggi's conjecture. Wulff shapes are the unique such sets.

Let $P$ be a parallelohedron in $\mathbb R^3$ with unit inradius. Bezdek's Rhombic Dodecahedral Conjecture. The surface area of $P$ is at least … which is the surface area of the r…

Let $P$ be a three-dimensional parallelohedron of unit volume, and let $\operatorname{surf}(P)$ denote its surface area. Regular truncated-octahedron conjecture. Among three-dimens…

Longest minimal length partition conjecture. Given $c\in(0,1)$, the set $\Omega$ maximizing $L(\Omega,c)$ under the constraint $|\Omega|=v_d$ is the ball. Given $n>1$ and…

Let $\phi$ be a norm of class $\mathrm{C}^\infty_+$, and let $\phi$-isoperimetric sets be isoperimetric sets for the norm $\phi$ in $\mathbb{H}^1$. Smooth regularity conjecture. Ev…

Let $0<\alpha<1$, let $\mu=w(x)dx$ be a strictly log-concave probability measure supported on all of $\mathbb{R}^n$, with centrally symmetric density $w(-x)=w(x)$, and let $E$ be a…

Let $\Omega\subset\mathbb{R}^n$ be convex, bounded, and centrally symmetric, and let $E$ be an isoperimetric subset of $\Omega$ with $|E|/|\Omega|=1/2$. Two hyperplane conjecture.…

Let $\mu$ be a finite, strictly log-concave measure on $\mathbb{R}^n$, and let $E$ be an isoperimetric set with volume fraction $0<\alpha<1/2$. Convex sandwich conjecture. There ar…

Let $\mu=e^{-V}dx$ be a strictly log-concave probability measure on $\mathbb{R}^n$, and let $E$ be an open isoperimetric subset with $\mu(E)=\alpha$. Lipschitz-interface conjecture…

Let $\Omega\subset\mathbb{R}^n$ be a bounded, convex domain, and let $E\subset\Omega$ be an open isoperimetric subset. Lipschitz-interface conjecture. The interface…

Let $H(\mathbb R)$ be the Heisenberg group with a Carnot–Carathéodory metric arising from a norm on the horizontal plane. In the sub-Riemannian case, Pansu's bubble set is the uniq…

Let $(H(\mathbb R),d_Q,\lambda)$ be the Heisenberg group equipped with Haar measure $\lambda$ and an arbitrary Carnot–Carathéodory metric $d_Q$ arising from a norm on the horizonta…

Let $H(\mathbb R)$ be the Heisenberg group with its sub-Riemannian Carnot–Carathéodory metric. Let $S$ be a surface enclosing a region $E$, let $\lambda$ denote Haar measure, and l…

Higher-dimensional double-bubble classification conjecture. A perimeter-minimizing double bubble is either

Let $S_g$ be a closed hyperbolic surface of genus $g$, and let $D_{\operatorname{inj}(S_g)}$ denote an embedded metric disk of radius equal to the injectivity radius of $S_g$. Schm…