HMMT 2012 G7

Let $S$ be the set of the points $\left(x_{1}, x_{2}, \ldots, x_{2012}\right)$ in 2012-dimensional space such that $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+$ $\left|x_{2012}\right| \leq 1$. Let $T$ be the set of points in 2012-dimensional space such that $\max _{i=1}^{2012}\left|x_{i}\right|=2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$ ?