The assertion is false. Let
$$
A_n=[-n,n]\times(0,n^{-2}].
$$
After filling bounded complementary components if necessary, this is a <compact H-hull> with diameter asymptotic to $2n$. The <half-plane capacity of a low rectangle> estimate, together with <scaling and translation of half-plane capacity>, gives
$$
\operatorname{hcap}(A_n)=O(n^{-1})\longrightarrow0.
$$
Thus unbounded diameter need not force unbounded capacity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 3 / c / i / Solution

 Solution

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