I have a list of $m$ affine inequalities in $n$ variables of the following form 


$$a\_1 x\_1 + \cdots + a\_n x\_n \leq c\_n$$


I would like to know whether there is any point on the unit sphere in $\mathbb R^n$ that satisfies all of them. Is there any easy way to check whether this is the case?


(Based on randomly choosing points, it seems like there are no solutions, but I'd like an algorithm that's up to journal standards of proof.)



Intersecting a convex polytope with the unit sphere
It was [proved](https://pdfs.semanticscholar.org/1e16/eff75b69f9e85593fd0641cccb2fae7ec066.pdf) by Freund and Orlin that the problem of checking whether a polytope specified by linear inequalities is not entirely contained in a ball specified by its centre and radius is NP-complete. 
And it is obvious that one can assume that the ball is a unit ball, for the corresponding transformation (shift+scaling) can be carried out in polynomial time.


Thus there is little hope for an efficient algorithm for the problem at hand.


