4 problems

Let $\mathfrak{A}_{co}$ be the class of non-empty open convex subsets of $\mathbb{R}^m$, and consider the variational problem … For $m=2$, the thinning-rectangle conjecture. The su…

Let $\mathfrak{A}_{co}$ be the class of non-empty open convex subsets of $\mathbb{R}^m$, and consider the variational problem … where $w_{\Omega}$ is the torsion function and…

Saint-Venant's conjecture. Among all such domains, the solution $u$ assumes its largest maximum when $\Omega$ is a ball.

Let $\Omega\subset\mathbb R^2$ be a bounded convex domain, and let $u_\Omega$ denote its torsion function. Define the shape functional $F$ by the quantities introduced in the paper…