9 problems

Buttazzo–Gavitońe conjecture. The sharp upper bound for $\mathcal F$ in arbitrary dimension is

Let $\Omega_0$ and $\Omega_1$ be open, bounded, centrally symmetric subsets of $\mathbb{R}^n$, and let $t\in[0,1]$. Write … The Gaussian torsional rigidity of a set $\Omega$ is den…

Equilateral triangle torsional-rigidity conjecture. The equilateral triangle of unit width minimizes the torsional rigidity among shapes having unit minimal width.

Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^n$. Let $X(t,x)$ be the flow defined by … where $\nu$ is the unit outer normal, $u(t)$ is the torsion function on the…

Let $N\in\mathbb N$ and let $\theta\in(0,\pi)$. Consider triangles of area $1$ having a fixed interior angle $\theta$, and let $\rho_N$ denote their Bergman $N$-polynomial content…

Let $\Omega$ be a simply connected Jordan region in the plane. For $N\in\mathbb N$, let $\rho_N(\Omega)$ denote the squared distance from $\bar z$ to the polynomials of degree at m…

Let $\Omega$ be a planar region, let $\rho_N(\Omega)$ be the squared distance from $\bar z$ to the polynomials of degree at most $N$ in $L^2(\Omega,dA)$, and call the corresponding…

Let $\Omega$ range over planar pentagons with a fixed area, and let $T(\Omega)$ denote their Torsional rigidity. Saint-Venant's conjecture. Among all pentagons of given area, the r…

Isosceles right triangle conjecture. Among all right triangles with area $A>0$, the one with largest torsional rigidity is the isosceles right triangle with area $A$.