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.