  We present a form of Schwarz's lemma for holomorphic maps between convex
domains $D_1$ and $D_2$. This result provides a lower bound on the distance
between the images of relatively compact subsets of $D_1$ and the boundary of
$D_2$. This is a natural improvement of an old estimate by Bernal-Gonz\'alez
that takes into account the geometry of $\partial{D_1}$. Using similar
techniques, we also provide a new estimate for the Kobayashi metric on bounded
convex domains.
