4 problems

Let $D\subset\mathbb{R}^d$ be a domain whose boundary $\partial D$ is homeomorphic to the unit sphere in $\mathbb{R}^d$. Higher-dimensional Pompeiu conjecture. The domain $D$ has t…

For a function satisfying the mean value property for circles of two radii $r_0$ and $r_1$, where $r_0/r_1$ avoids the finite exceptional set determined by the relevant Bessel-func…

Higher-dimensional decomposition conjecture. For every $D\in\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})$,

Higher-dimensional analytic-content equality conjecture. The lower bound is attained only for balls and spherical shells.