On the roots of the Wills functional

We investigate the roots of Wills polynomials of convex bodies. We study the structure, showing that the set of roots in the upper half plane is a convex cone, monotonous with respect to the dimension. In particular, we give its precise description for dimensions n=2,3. We also show that for n≤7 this cone is completely contained in the (open) left half plane, which is not true in dimensions ≥14. Moreover, we study the size of the roots of the Wills polynomial, bounding them in terms of functionals like the in- and circumradius of the set. We also relate the roots of the Steiner and the Wills polynomials.