A geometric law of iterated logarithm for Brownian motions

Let {ω(t), t∈[0, 1]} be the standard d-dimensional Brownian motion. Set C(ω)={ω(s), O≤s≤t}(t∈[O, 1]) and call {C, t∈[0, 1]} the Brownian convex hull. Early in 1956, Levy got the following famous results: lim sup(2tloglog(l/t))V(C)=m, where V(·) denotes the volume functional and mis a non-trivial constant. In recent years,much attention has been re-paid to the study of the Brownian convex hull since the