Half-plane capacity is bounded by squared diameter
= Half-plane capacity is bounded by squared diameter
There is a universal constant $C$ such that every compact H-hull satisfies
$$
\operatorname{hcap}(A)\leq C\operatorname{diam}(A)^2.
$$
Translate and scale so the hull lies in a unit half-disc, then use the <Brownian representation of half-plane capacity> and the $O(y^{-1})$ <harmonic measure> of that half-disc as viewed from $iy$.