The infimum is zero; take $x$ a standard basis vector, and $w$ as close to that as you like.
Now that it appears that $w$ is given, and that $x$ is required to have at least one positive and one negative entry (I'll skip over the difficulty at $n=1$), it seems one can always make the angle smaller by (assuming wlog the initial angle acute) moving a negative entry of $x$ closer to $0$, and rescaling to a unit vector. So for the infimum we may suppose all entries of $x$ nonnegative with at least one equal to $0$. But then, fixing the set of positions where $x$ has coordinate $0$ by projecting $w$ onto the subspace with coordinates $0$ in those positions, one sees that one can do no better for $x$ than to choose its remaining coordinates equal (up tu a scalar) to those of $w$. I think it is not hard to see that the smallest angle is obtained by selecting the coordinate of minimal value in $w$ and making that zero (or infinitesimally negative in the original problem) in $x$, and rescaling.