  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 $K$ does 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 Jer\'onimo-Castro about the homothetic convex hull function, and
prove a homothetic variant of the translative constant volume property
conjecture for $3$-dimensional convex polyhedra. We also apply our results to
describe the properties of the illumination bodies of convex bodies.
