# Minimize the maximum inner product with vectors in a given set

Given a set $$S$$ of non-negative unit vectors in $$\mathbb R_+^n$$, find a non-negative unit vector $$x$$ such that the largest inner product of $$x$$ and a vector $$v \in S$$ is minimized. That is, $$\min_{x\in \mathbb R_+^n,\|x\|_2=1}\max_{v\in S} x^Tv.$$

It seems like a quite fundamental problem in computational geometry. Has this problem been considered in the literature?

It can be formulated as an infinity norm minimization problem, which can in turn be expressed as a quadratically constrained LP. If the rows of matrix $$A$$ are the vectors in $$S$$, we seek \begin{align} &&\min_x\|Ax\|_\infty \ \rm{s.t.} && x^Tx=1 \ && x\geq 0. \end{align} But the quadratic constraint is non-convex, so this is not very encouraging.