6 problems

Let $n\geq 2$, let $q(n)\leq n-2$, and let $K\subset\mathbb R^n$ be compact. Write $\operatorname{diam}(K)$ for its diameter and $\operatorname{Cap}_q(K)$ for its Riesz capacity. I…

Let $K\subset\mathbb{R}^n$ be a convex body, let $idq(K)=\frac{\operatorname{Vol}(K)}{\operatorname{Diam}(K)^n}$ be its isodiametric quotient, and let $T$ be a linear bijection of…

Dvoretzky-Rogers equality conjecture. For every $n\in\mathbb{N}$,

Makai Jr.'s conjecture. For every $K\in\mathcal{K}^n$ there is a linear transformation $A\in\operatorname{GL}_n(\mathbb{R})$ such that

Higher-dimensional Alexandrov conjecture. The inequality

Alexandrov's isodiametric conjecture. For any such surface,