Say we want to find a sparse approximate minimizer to the function $f(x) : \mathbb{R}^d \to \mathbb{R}$. Then in line with the work in the field of compressed sensing, we can instead minimize $$f(x) + \lambda \|x\|_1$$ as a convex proxy/relaxation to the desired sparsity constraint on $\|x\|_0$.
But often times optimization problems are actually already restricted to an $\ell^1$ ball. For example, say we are optimizing over the space of discrete probability distributions on $d$ elements, which can be represented as \begin{align*} \text{minimize: } & f(x)\\ \text{subj. to: } & x \succeq 0\\ & \langle \boldsymbol{1}, x \rangle = 1. \end{align*} Call that problem (P). Then since any feasible solution to (P) has $\|x\|_1 = 1$ exactly, adding the objective penalty $\lambda\|x\|_1$ cannot change the optimal solution of the problem.
Question: Is there a way to add a convex penalty / constraint to (P) so that the resulting problem (P') has a sparse solution?