Revisiting the approximate Carathéodory problem via the Frank-Wolfe algorithm

The approximate Carathéodory theorem states that given a compact convex set $${\mathcal {C}}\subset {\mathbb {R}}^n$$ and $$p\in [2,+\infty [$$ , each point $$x^*\in {\mathcal {C}}$$ can be approximated to $$\epsilon $$ -accuracy in the $$\ell _p$$ -norm as combination of {O}}(pD_p^2/\epsilon ^2)$$ vertices where $$D_p$$ is diameter -norm. A solution satisfying these properties built using probabilistic arguments or by applying mirror descent dual problem. We revisit problem solving primal via Frank-Wolfe algorithm, providing simplified analysis leading an efficient practical method. Furthermore, improved cardinality bounds are derived naturally existing convergence rates algorithm different scenarios, when $$x^*$$ interior subset with small diameter, uniformly convex. also propose [1,2[\cup \{+\infty \}$$ nonsmooth variant algorithm. Lastly, we address finding sparse projections onto -norm, [1,+\infty ]$$ .