27 problems

Let $(M^3,g)$ be an asymptotically flat Riemannian 3-manifold in the asymptotically Schwarzschild setting. Consider the volume-preserving stable constant mean curvature spheres con…

Let $P$ be a stationary regular partition of $\mathbb{R}^n$. A partition is strictly area-minimizing in its tangent cones when all tangent cones of $P$ have this property. It is lo…

Graham's conjecture. For every even $n\ge6$, the diameter graph of every optimal $n$-gon consists of an $(n-1)$-cycle together with one additional edge emanating from one of the cy…

Isoperimetric optimality conjecture. The 120-cell has the least volume among all 4-polytopes of unit in-radius having 120 cells.

Let $\Om\subset\mathbb{R}^3$ be a strictly convex body, and let $\Sg$ be an isoperimetric hypersurface inside $\Om$. Ros's disk conjecture. The hypersurface $\Sg$ should be homeomo…

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be t…

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be t…

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…

Sullivan's conjecture. For $M\le d+1$, isoperimetric $M$-clusters coincide with standard clusters.

A spider web is the tensegrity structure described in the paper, enclosing a sphere in the semi-global setting. The circumscribed condition is the requirement that the polytope be…

Generalized-curvature conjecture. The generalized curvature $kappa_psi$ and the Lagrange multiplier $lambda$ are equivalent, since $lambda$ can be interpreted as weighted bubble pr…

Connectedness conjecture. Isoperimetric clusters have simply connected chambers.

Let $\mathbf{R}^3$ carry the radial density $r^p$. A standard double bubble is the usual two-region double bubble bounded by spherical surfaces meeting along a singular circle at…

Hauswirth's three-phase conjecture. The isoperimetric profile of $T^2\times\mathbb R$ is composed of three parts: for small volumes, isoperimetric regions are spheres $B^3_R$; for…

Transition-volume equality conjecture. In fact,

Three-phase profile conjecture. The precise isoperimetric profile of $(T^2\times {\mathbb R},h_2+dt^2)$ should be generated by regions such as spheres $B^3_R$, cylinders…

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $I_{\mathbb{P}^{n+1}(\mathbb{K})}$ denote the isoperimetric profile of $\mathbb{P}^{n+1}(\mathbb{K})$, defined by ……

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, and let $\mathbb{P}^{n+1}(\mathbb{K})$ be the corresponding projective space. A projective subspace…

Let $\mathbb{H}^n$ be hyperbolic space with a smooth, radial, log-convex density, and let spheres about the origin mean geodesic spheres centered at the origin. The log-convex dens…

Let $f(x,y)=e^{\psi(x,y)}$ be the normalized sum of two Gaussian densities with the same variance and different centers. The line connecting the two centers determines a perpendicu…

Let $\mathbb{R}^n$ carry a finite sum of Gaussian measures whose centers lie on the $x$-axis. A half-space is vertical when its boundary is a hyperplane perpendicular to that axis.…

A standard simplex partition of $\mathbb{R}^n$ is obtained from a regular simplex with center $P$ by taking, for each facet, the cone over that facet with common base point $P$. St…

Let $M$ be an $n$-dimensional non-positively curved manifold. The isoperimetric conjecture asserts that $M$ satisfies the isoperimetric inequality, namely that among Euclidean $n$-…

Radial-density hypersphere conjecture. In $\mathbb{R}^{n}$ with density $r^{p}$, hyperspheres through the origin are uniquely isoperimetric.