8 problems

Let $K\subset\mathbb R^2$ be compact. The logarithmic capacity $\operatorname{Cap}_0(K)$ and Newtonian capacity $\operatorname{Cap}_1(K)$ are the corresponding Riesz capacities. Pó…

Let $n\geq1$ and let $p\in(n-2,n)$. For compact sets $A,B\subset\mathbb R^n$ and $\lambda\in[0,1]$, write … Let $\operatorname{Cap}_p$ denote the Riesz $p$-capacity. Novaga–Ruffini…

Let $n\geq 2$, let $q<p<0$, 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 corres…

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 $n\geq 2$, 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 $n\geq 1$, let $p<q$, and let $K\subset\mathbb R^n$ be compact with positive $p$-capacity. The quantities $\operatorname{Cap}_q(K)$, $\operatorname{Cap}_p(K)$, and…

Equality conjecture. If $A$ is a strictly self-similar fractal with dimension $d$, then

Moments of Riesz $p$-equilibrium measures are minimal for the ball. For every $q\in(0,\infty)$,