4 problems

Let $\mathbf{K}$ and $\mathbf{C}$ be convex bodies in $\mathbb{E}^d$, with $d\geq 2$, and let $m$ be a positive integer. For each body, let $r_{\mathbf{C}}(\cdot,m)$ denote its $m$…

Let $\Omega\subset\mathbb R^d$ be a domain of finite measure with inradius $\varrho$, and let $\lambda_{1}^{(m)}(\Omega)$ denote its first polyharmonic eigenvalue. Inradius asympto…

Let a two-dimensional $\operatorname{CBB}(\kappa)$ space contain $\lambda$-convex domains homeomorphic to a disk, and fix their area. Area-constrained inradius conjecture. Among al…

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, and let $p_t(x)$ denote its heat-content quantity. Heat-content inradius conjecture. If there is a constant $c_1>0$ such that…