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

The assertion is false. LetAfter 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, givesThus unbounded diameter need not force unbounded capacity.

The assertion is true. The standard estimate half-plane capacity is bounded by squared diameter givesConsequently $\mathrm{hcap}(A_n)\to \infty$ forces $\mathrm{diam}(A_n)\to \infty$.