Two remarks on remotality

We prove that there exists a weakly closed and bounded subset $E$ of $c_0$ which is not remotal from 0, and such that $\overline{\mathrm{co}}\left(E\right)$ is remotal from 0. This answers a question of M. Martín and T.S.S.R.K. Rao. We also present a simple proof of the fact that in every non-reflexive Banach space there exists a closed convex bounded set which is not remotal.