Curvature of the convex hull of planar Brownian motion near its minimum point

Abstract

Let f be a (random) real-valued function whose graph represents the boundary of the convex hull of planar Brownian motion run until time 1 near its lowest point in a coordinate system so that f is non-negative and f(0) = 0. The ratio of f(x) and |x|/|log |x|| oscillates near 0 between 0 and infinity a.s.