Minimax of an n-dimensional Brownian motion

For some absolute constants $c$, $n_0$ and any $n\geq n_0$, we show that with probability close to one the convex hull of the $n$-dimensional Brownian motion ${\rm conv}\{BM_n(t):\, t\in[1,2^{cn}]\}$ does not contain the origin. The result can be interpreted as an estimate of the minimax of the Gaussian process $\{ \langle \bar{u},BM_n(t)\rangle,\, \bar{u}\in S^{n-1},\, t\in [1,2^{cn}]\}$.