I have a list of affine inequalities in $n$ variables, i.e. inequalities of the 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.)

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.)