Let $R_r=[-r,r]\times(0,1]$ and set
$$
A_n=n^{-2}R_{n^3}
=[-n,n]\times(0,n^{-2}].
$$
The <scaling and translation of half-plane capacity> and part i give
$$
\operatorname{hcap}(A_n)
=n^{-4}\operatorname{hcap}(R_{n^3})
\leq\frac Cn\longrightarrow0.
$$
On the other hand,
$$
\operatorname{diam}(A_n)=\sqrt{4n^2+n^{-4}}\longrightarrow\infty.
$$
Thus very long, very shallow hulls can have vanishing half-plane capacity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 1 / d / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-203/1/d/ii/solution