9 problems

Let a planimeter of length $\l$ trace the boundary of a region $\O$ with area $A_\O$. When the traced path is a closed tractrix, the planimeter makes a complete rotation. Menzin's…

Stadium conjecture. The stadium $S$ that minimizes $\delta/\lambda^2$ also minimizes the functional $\delta/\lambda_0^2$. This is a conjecture about the shape of the optimal convex…

Let $\Omega\subset\mathbb{R}^2$ be a compact connected set, and let $\delta(\Omega)$ denote its isoperimetric deficit and $\lambda_0(\Omega)$ its barycentric asymmetry. Dumbbell co…

Let $\Omega\subset\mathbb{R}^2$ be a compact convex set, and let $\delta(\Omega)$ denote its isoperimetric deficit and $\lambda_0(\Omega)$ its barycentric asymmetry. A stadium is t…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ with $N\geq 3$, volume $V$, and boundary whose finitely many smooth components have total surface area $P$. Let $r_{\Omega}>0$…

Analytic-content extremal-domain conjecture.

Let $\mathsf{M}_0$ be the mask-shaped set described in the paper, with parameters $\alpha$, $\theta$, and $x_0$. The quantitative isoperimetric functional is denoted by…

Let $\mathcal{F}$ be the quantitative isoperimetric functional considered in the paper, and let $\Omega_0$ denote an optimal set for $\mathcal{F}$. Suppose that $\Omega_0$ is conne…

Let $K=B_1$ be the Euclidean unit ball, let $E\subset\mathbb{R}^n$ be a measurable set of finite positive measure, and define its isoperimetric deficit and asymmetry index by … … T…