3 problems

Let $S$ be an orthocentric simplex in any dimension, with facet center ${\cal F}$ and incenter ${\cal I}$. Regularity conjecture. If ${\cal F}={\cal I}$, then $S$ must be regular.…

Let $n\geq 3$, let $p<q\leq -2$, and let $K\subset\mathbb R^n$ be compact with more than one point. The quantities $\operatorname{Cap}_p(K)$ and $\operatorname{Cap}_q(K)$ are the c…

Let a Euclidean $n$-simplex $\overline{\mathsf{ABC} \cdots}$ have $n(n+1)$ contact $(n-1)$-simplices, occurring in congruent pairs, and let…