7 problems

Let $A$ and $B$ be homothetic regions, and let $\tau$ and $\delta$ denote the parameters measuring the relative size and Brunn–Minkowski deficit introduced in the paper. Let…

Let $\Omega\subset\mathbb{R}^d$ be a domain, and let the Fraenkel asymmetry of $\Omega$ be denoted by $\mathcal{A}(\Omega)$ as in. Let $\lambda_1(\Omega)$ be the first Dirichlet ei…

Let $u$ solve the torsion problem on a domain $\Omega$ with boundary $\Gamma$, and let $u_\nu$ denote its normal derivative on $\Gamma$. Quantitative Serrin stability bounds measur…

Let $d$ be an integer with $|d|\geq 2$. For a map $u:\mathbb{R}^2\to\mathbb{S}^2$, let $\mathcal{E}(u)$ denote its energy and $\deg(u)$ its degree. Let $\mathcal{A}_d$ be the class…

Let $u\in\dot H^{1/2}(\mathbb{R};\mathbb{S})$ be a half-harmonic map of degree $d>0$, and let $\psi_{\vartheta,\vec{\alpha}}$ denote a degree-$d$ half-harmonic map. Quantitative st…

Quantitative stability conjecture. There exists $\eta>0$ such that, for every $V\in\mathcal M(\Omega)$,

Let $E\subset\mathbb{R}^n$ be an open set with Lipschitz boundary and Gaussian measure $\gamma(E)=\phi(s)$, where … Its Gaussian perimeter is … and define the Gaussian isoperimetri…