How to determine whether there exist vectors having positive inner products with a given set of vectors?

Question

Given $m$ real-number vectors $\{\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_m\}$ with dimension $n$, is there a method to determine whether there exist $n$-dimensional vectors $\vec{x}$ such that the inner product $\vec{x}_i^T\vec{x}>0$ for all $i=1,2,\cdots,m$ ?

Answer 1

Solve the linear program $\min_{x,\alpha} \{ \alpha | x_k^T x \le \alpha \}$. If $\alpha <0$ then yes, otherwise no.

Alternatively, find the nearest point $d$ to the origin in the set $\operatorname{co} \{ x_k \}$. If $d=0$ then no such direction exists, otherwise $x_k^Td >0$ for all $k$.

Note that these are, at some level, dual approaches.