Question 39
Let $S$ be the set of the points $(x_{1}, x_{2}, \ldots, x_{2012})$ in 2012-dimensional space such that $|x_{1}| + |x_{2}| + \cdots + |x_{2012}| \leq 1$. Let $T$ be the set of points in 2012-dimensional space such that $\max_{i=1}^{2012} |x_{i}| = 2$. 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 $S$?