Minimax of an n-dimensional Brownian motion

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