Learning convex bodies is hard

We show that learning a convex body in $\RR^d$, given random samples from the body, requires $2^{\Omega(\sqrt{d/\eps})}$ samples. By learning a convex body we mean finding a set having at most $\eps$ relative symmetric difference with the input body. To prove the lower bound we construct a hard to learn family of convex bodies. Our construction of this family is very simple and based on error correcting codes.

💡 Research Summary

The paper investigates the fundamental sample‑complexity limits of learning an unknown convex body in $\mathbb{R}^d$ when the learner is given only independent random points drawn uniformly from the body. “Learning” is defined in the standard statistical‑geometry sense: the algorithm must output a set $\tilde K$ whose relative symmetric difference with the true body $K$ satisfies \