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.