10 problems

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

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…

Consider $\mathbf{R}^n$ with density $r^p$ for positive $p$. An isoperimetric double bubble conjecture asserts that the best double bubble is the Euclidean standard double bubble,…

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:

Transitional-angle conjecture. The circle is isoperimetric as long as it has nonnegative second variation, and the semicircle is isoperimetric for every angle greater than…

In $\mathbb{R}^{N}$, let the density be smooth, radial, and log-convex. Bayle's conjecture. Balls centered at the origin are isoperimetric. The conjecture is motivated by a propose…

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), (…

Consider the plane $\mathbb{R}^2$ with density … An isoperimetric boundary is the boundary of a region minimizing weighted perimeter among regions with prescribed weighted area. Ve…