A Geometric Interpretation of Half-Plane Capacity

Let A be a bounded, relatively closed subset of the upper half plane H whose complement in H is simply connected. If Bt is a standard complex Brownian motion and τA = inf{t ≥ 0 : Bt 6∈ H \A}, the half-plane capacity hcap(A) is defined as hcap(A) := lim y→∞ y E [Im(BτA)] . This quantity arises in the study of Schramm-Loewner Evolutions (SLE). In this note, we show that hcap(A) is comparable to a more geometric quantity hsiz(A) that we define to be the 2dimensional Lebesgue measure of the union of all balls tangent to R whose centers belong to A. Our main result is that 1 66 hsiz(A) < hcap(A) ≤ 7 2π hsiz(A).