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

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- Solution i

Solution

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

2 n

. The half-plane capacity of a low rectangle estimate, together with scaling and translation of half-plane capacity, gives

hcap (A_{n}) = O (n^{- 1}) ⟶ 0.

Thus unbounded diameter need not force unbounded capacity.

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