Consider an optimization problem over infinite variables:

$$
\begin{align}
\min_{x}~& {\left\lVert{x}\right\rVert }_p
\\
\text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\dots,N
\end{align}
$$
where $N\in\mathbb{N}$, $x$ and $\left\{a_n\right\}_{n=1}^{N}$ are all **infinite-length** vectors,
the $a$'s are constant vectors, and ${\left\lVert{\cdot}\right\rVert }_p$ is the $p$-norm.

**Clarification on the constraints** (edited): we can assume each entry of the constraint vectors is bounded by a constant $r>0$, that is $\forall{n\in\left[{N}\right]},i: \left\lvert{a_n\left({i}\right)}\right\rvert \le r$. Unfortunately, we *cannot* assume that these entire vectors $\left\{{a_n}\right\}_{n=1}^{N}$
are bounded under some norms.

**Prove:** *if the minimum is attainable*, then there exists an optimal solution $x^*$ whose support, i.e. $\text{supp}\left(x^*\right)$, is finite (where the support of a vector is its non-zero entries).

I am especially interested in cases where $0<p<1$, **when the objective function is no longer convex**.

When $p=1$, there are some known proofs (*e.g.* on Wei 2018), but as far as I understand they all use the **convexity** of the $p$-norm when $p\ge 1$, *e.g.* to apply strong duality to the dual problem and show that there are optimal solutions with a support of at most $N$.

I started reading about quasi-convex optimization (since $p$-norms for $p\in\left[0,1\right]$ are quasi-convex), but I was thinking maybe there is a simple solution I am missing out.

**Update**: since it is already known for $p=1$, one could (at least practically) expect the sparsity would only improve for lower values of $p$. So if there are some theoretical results in that spirit, they could be relevant.

Any help or directions will be highly appreciated.

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