Research Problems in Function Theory — Problem 3.18

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a Borel set $E\subset\mathbb R^n$ of finite positive volume $V$, let $B_V$ be any ball of volume $V$, and define \[\delta(E)=\inf_{|B|=V}\frac{|B\setminus E|}{V}.\] For $n\geq3$, capacity is Newtonian capacity, defined by the infimum of the probability-measure energy with kernel $|x-y|^{2-n}$ and then taking its reciprocal. In $n=2$, use logarithmic capacity $\exp(-\inf_\mu\iint\log(1/|x-y|)\,d\mu(x)\,d\mu(y))$. Measures are supported on $E$, with the usual capacitable-set extension; infinite capacity is allowed. Put $C_{n,V}=\operatorname{cap}(B_V)$. The source update gives an improvement of order $\delta^{n+1}$ for general planar sets and for convex sets in dimensions $n\geq3$.

Statement recorded in the supplied source.

For each $n\geq2$, is there a function $\Phi_n:(0,1]\to(0,\infty)$, depending only on $n$, such that every such $E$ with $\delta(E)>0$ satisfies \[\operatorname{cap}(E)\geq C_{n,V}\left(1+\Phi_n(\delta(E))\right)?\] Determine a quantitative positive improvement over the ball's capacity in terms of this asymmetry. [S2]

Short English statement

Does every set with a definite departure from all equal-volume balls have capacity larger than the ball by a quantitatively positive amount?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303018 (AMR-022-3018). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 3.18 and Update 3.18. https://arxiv.org/abs/1809.07200.