  We investigate structural properties of the cone of roots of relative Steiner
polynomials of convex bodies. We prove that they are closed, monotonous with
respect to the dimension, and that they cover the whole upper half-plane,
except the positive real axis, when the dimension tends to infinity. In
particular, it turns out that relative Steiner polynomials are stable
polynomials if and only if the dimension is $\leq 9$. Moreover, pairs of convex
bodies whose relative Steiner polynomial has a complex root on the boundary of
such a cone have to satisfy some Aleksandrov-Fenchel inequality with equality.
An essential tool for the proofs of the results is the characterization of
Steiner polynomials via ultra-logconcave sequences.
