Say I had s vectors in $R^4$ whose entries were in the set of non-negative integers. Let $\bar{v}$ be the average of these vectors. How can you prove that there exists a subset of $\frac{s}{2} + 2$ of these vectors such that their sum is greater than $\frac{s}{2}\times \bar{v}$ in all four dimensions? I was able to prove this in the case of R^2 but my approach didn't generalize to higher dimensions, could anyone give me an idea of how to get started?
Sorry in advance English is my second language