  The theory of coconvex bodies was formalized by A.~Khovanski{\u\i} and
V.~Timorin in \cite{KT}. It has fascinating relations with the classical theory
of convex bodies, as well as applications to Lorentzian geometry. In a recent
preprint \cite{schnei2}, R.~Schneider proved a result that implies a reversed
Brunn--Minkowski inequality for coconvex bodies, with description of equality
case. In this note we show that this latter result is an immediate consequence
of a more general result, namely that the volume of coconvex bodies is strictly
convex. This result itself follows from a classical elementary result about the
concavity of the volume of convex bodies inscribed in the same cylinder.
