How to minimize $L_1$ norm in a subspace when $L_2$ norm is fixed?

Question

I want to solve the following optimization problem $$\text{minimize}~~\lVert \vec{x} \rVert_1$$ $$\text{subject to}~~\lVert \vec{x} \rVert_2 = 1$$ $$\text{and}~~A \vec{x} = \vec{0}$$ where $\vec{x} \in \mathbb R^n$, $\lVert \vec{x} \rVert_1 = \sum_{i=1}^n |x_i|$, $\lVert \vec{x} \rVert_2 = (\sum_{i=1}^n |x_i|^2)^{1/2}$, $A \in \mathbb R^{m\times n}$ and $m<n$. This is not a convex optimization problem because the domain of $\vec{x}$ is not convex. I wonder if there is any method to solve this problem, or even some method in special cases would be good.