Highest sparsity vector in a vector subspace

Question

I am aware of some work being done on recovering a sparse vector from a subspace over $\mathbb{R}$ (here). Has any work been done on finding the highest sparsity ($\min \{ \|x\|_0 : x\in \mathbb{S} \}$) of a vector subspace $\mathbb{S}$, and/or the number of such 'highest sparse' vectors?