Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex has maximum overlap with $\Delta_n$. Formally:
$$ \max_{y \in \mathbb{R}^n}\ \text{vol} (\Delta_n \cap (-\Delta_n+y)) $$
I wrote the volume as an integral of indicator function , but I don't know how to maximize that integral in n-dimensions.