8 problems

Exponent-two conjecture. The exponent $2$ should be the optimal exponent in a quantitative estimate in terms of the Fraenkel asymmetry, instead of the exponent $3$ in the displayed…

Quadratic-exponent optimality conjecture. The quadratic exponent is optimal in these quantitative inequalities of spectral type.

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

Sharp quantitative Brunn–Minkowski stability conjecture. For arbitrary sets $E$ and $F$, the optimal exponent of $\alpha(E,F)$ in a quantitative stability inequality for the Brunn–…

Bhattacharya–Weitsman–Nadirashvili conjecture. There exists a dimensional constant $\sigma>0$ such that

Let $E\in\mathcal S^2$ be a planar set, with perimeter deficit $\delta P(E)$ and Fraenkel asymmetry $\alpha(E)$. Hall's conjecture. One has … and the constant…