I have to maximize the function $g(y) = inf_i{\|y - x_i\|_2}$ subject to $y\in B_0(1)\subset\mathbb{R}^n$.
Then I thought that maybe there is an averaging or mollifying of the functions (using partition of unity of some sort) that might do the trick and the $y$ that maximizes this average could be the same as the $y$ that maximizes $g(y)$. Does anybody know if that is the case? What is done in general in this situation?
I also thought about finding the Voronoi diagram and solving the problem in each cell (which is convex in each cell), but it does not seem very elegant to me.
Thank you very much.