The functionalW(A), defined by J. M. Wills for all convex bodiesA in then-dimensional Euclidean space, is a linear expression\(W(A) = \sum\nolimits_0^n {\left( {\begin{array}{*{20}c} n \\ v \\ \end{array} } \right)\frac{1}{{\omega _v }}W_v (A)} \), whereWv is the “vth Quermas-integral” andωv stands for the volume of thev-dimensional unit ball. In the present note several remarkable properties ofW(A) are explained. Especially, we have Wills's conjectureG(A)≤W(A), whereG(A) is the number of points of then-dimensional unit lattice contained inA.