  The aim of this note is to investigate the properties of the convex-hull and
the homothetic convex-hull functions of a convex body $K$ in Euclidean
$n$-space, defined as the volume of the union of $K$ and one of its translates,
and the volume of $K$ and a translate of a homothetic copy of $K$,
respectively, as functions of the translation vector. In particular, we prove
that the convex-hull function of the body $K$ do not determine $K$.
Furthermore, we prove the equivalence of the polar projection body problem
raised by Petty, and a conjecture of G.Horv\'ath and L\'angi about translative
constant volume property of convex bodies. We give a short proof of some
theorems of Castro about the homothetic convex-hull function, and prove a
homothetic variant of the translative constant volume property conjecture for
$3$-dimensional convex polyhedra.
