Haar null closed and convex sets in separable Banach spaces

Haar null sets were introduced by Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, Darji defined a categorical version of Haar null sets, namely Haar meagre sets. The present paper aims to show that, whenever [Formula: see text] is a closed, convex subset of a separable Banach space, [Formula: see text] is Haar null if and only if [Formula: see text] is Haar meagre. We then use this fact to improve a theorem of Matoušková 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.