  The Wills functional $\mathcal{W}(K)$ of a convex body $K$, defined as the
sum of its intrinsic volumes $\mathrm{V}_i(K)$, turns out to have many
interesting applications and properties. In this paper we make profit of the
fact that it can be represented as the integral of a log-concave function,
which, furthermore, is the Asplund product of other two log-concave functions,
and obtain new properties of the Wills functional (indeed, we will work in a
more general setting). Among others, we get upper bounds for $\mathcal{W}(K)$
in terms of the volume of $K$, as well as Brunn-Minkowski and Rogers-Shephard
type inequalities for this functional. We also show that the cube of
edge-length 2 maximizes $\mathcal{W}(K)$ among all $0$-symmetric convex bodies
in John position, and we reprove the well-known McMullen inequality
$\mathcal{W}(K)\leq e^{\mathrm{V}_1(K)}$ using a different approach.
