12 problems

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$, and let a cubical subset be a compact subset of $[0,1]^n$ whose boundary is contained in finitely many hyperplanes with outer norm…

Let $[0,1]^n$ be the unit cube in $\mathbb{R}^n$ with the Euclidean metric. For a measurable set $U\subset [0,1]^n$ of locally finite perimeter, let $\operatorname{Vol}(U)$ denote…

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 $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, 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…

Log-Convex Density Conjecture. In $\mathbb{R}^{n}$ with radial log-convex simple density, balls about the origin are isoperimetric for every volume.

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

Plane radial-density conjecture. For $k\in[0,\infty)$, the following curves are isoperimetric: