6 problems

Let $M^n$ carry the metric … and density potential $\phi=x_1+x_2+\cdots+x_n$. Compute the holonomy Lie algebra at the origin $(0,0,\ldots,0)$. Strictly upper triangular holonomy co…

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:

Let $\mathbb{R}^2$ carry the density … An isoperimetric region minimizes weighted perimeter among regions of prescribed weighted volume. Horizontal-or-vertical-line conjecture. Eve…

Let $\mathbb{R}^2$ have the strip density, equal to $1$ in the strip $\{|y|\leq 1\}$ and equal to $\lambda>1$ outside it. The relevant candidate sets are classified as types (i), (…