The expected perimeter of the convex hull of planar Brownian motion exiting the unit disk equals 2π²(Si(π) − Si(π/2)) − 2π ≈ 3.2148, derived via conformal mapping of a truncated disk to the upper half-plane.
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Expected perimeter and area of the convex hull of planar Brownian motion stopped upon exiting the unit disk
The expected perimeter of the convex hull of planar Brownian motion exiting the unit disk equals 2π²(Si(π) − Si(π/2)) − 2π ≈ 3.2148, derived via conformal mapping of a truncated disk to the upper half-plane.