  Haar null sets were introduced by J.P.R. Christensen in 1972 to extend the
notion of sets with zero Haar measure to nonlocally compact Polish groups. In
2013, U.B. Darij defined a categorical version of Haar null sets, which he
named Haar meagre sets. The present paper aims to show that, whenever $C$ is a
closed, convex subset of a separable Banach space, $C$ is Haar null if and only
if $C$ is Haar meagre. We then use this fact to improve a theorem of E.
Matou\v{s}kov\'{a} and to solve a conjecture proposed by Esterle, Matheron and
Moreau. Finally, we apply the main theorem to find a characterisation of
separable Banach lattices whose positive cone is not Haar null.
