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We connect this question to a problem of estimating the probability that theimage of certain random matrices does not intersect with a subset of the unitsphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownianmotion is related to Gordon's escape theorem dealing with standard Gaussianmatrices. The approach allows us to prove that with high probability, the$\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order$n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we extend the"escape" phenomenon to a broad class of random matrices; as an application, weshow that $e^{Cn}$ steps are sufficient for the standard walk on $\mathbb{Z}^n$to absorb the origin into its convex hull with a high probability.

When does a discrete-time random walk in $\mathbb{R}^n$ absorb the origin into its convex hull?