Consider the classical problem of finding the smallest n such that any d-dimensional subspace of Lp(0,1) can be (1+?)-embedded into (Rn,||.||p). We show that when p≥1 is not an even integer and d≥C∗log(1/?), it is necessary that n≥C′/(?2polylog(1/?)), which is the optimal dependence on eps up to polylogarithmic factors, extending the previously known best eps-dependence of 1/?2−o(1) for p=1 to a general p. The proof uses techniques in communication complexity by studying the following ?p-subspace sketch problem. There are two players Alice and Bob. Alice has the subspace and sends a message to Bob such that Bob can approximate the ellp norm of any vector in the subspace. We show that the message length needs to be at least d/(?2polylog(1/?)) bits, from which the embedding dimension lower bound follows.