25 problems

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 simply connected domain representing the cross-section of a beam, with prescribed area. Its torsional rigidity $T(\Omega)$ is defined by the associated Rayleigh q…

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

Let $N\geq 2$ and let $\Omega\subseteq\mathbb R^N$ be a bounded convex open set. Denote by $T_2(\Omega)$ its torsional rigidity, by $|\Omega|$ its volume, and by $P(\Omega)$ its pe…

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…

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…

Henrot–Lucardesi–Philippin conjecture. Every convex planar domain satisfies

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 $\mathcal{A}$ be the admissible class of domains, let $T(\Omega)$ denote the torsional rigidity, let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue, and let $B$ be the b…

Pólya–Szegő polygonal Saint-Venant conjecture. The unique solution is the regular polygon with $n$ sides and area $\pi$.

Berger's conjecture. The constant $\frac{\pi^2}{8}$ is the optimal lower bound for the product $\lambda_1\|v\|_\infty$. The value is attained for path graphs with one or two Dirich…

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…

Let $\Omega\subset\mathbb{R}^n$ be a bounded, open, convex set, and let $T_p(\Omega)$, $P(\Omega)$, and $|\Omega|$ denote its relevant torsional quantity, perimeter, and volume. Fo…

Sharp-exponent conjecture. The sharp exponent in this quantitative width estimate is $1$, as in the planar case. The source establishes an estimate with exponent $n-1$ and conjectu…

Let $\Omega\subset\mathbb{R}^n$ be a bounded, open, convex set, and let … Here $P(\Omega)$ is the perimeter, $T(\Omega)$ is the torsional rigidity, and $|\Omega|$ is the Lebesgue m…

St. Venant's conjecture. Among simply connected domains of a given volume $v$, the torsional rigidity $T(\Omega)$ is maximized by the round ball.

Let $\Omega$ range over domains of fixed area, and let $u$ be the torsion function on $\Omega$, so that its torsional rigidity is $\int_{\Omega}u(x)\,dx$. Saint Venant's conjecture…

For a bounded domain $\Omega\subset\mathbb{R}^N$, let $T_p(\Omega)$ denote its anisotropic $p$-torsional rigidity, let $M(\Omega)$ denote the relevant geometric quantity, and defin…

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 $A$.

Isosceles right triangle conjecture. Among right triangles of area $1$, the one with largest torsional rigidity is the isosceles right triangle.

Polya's conjecture. The $n$-gon of area $1$ with maximal torsional rigidity is the regular $n$-gon of area $1$.

Disk extremality conjecture. Among regions with fixed area, disks have maximal torsional rigidity.

Let $\mathcal B(t)$ be the Brownian boundary at time $t$ in dimension $m=2$, and let $\lambda_1(\mathcal B(t))$ denote its principal Dirichlet eigenvalue. Write $\mathbb E_0$ for e…