A characterization of dual quermassintegrals and the roots of dual Steiner polynomials

Abstract

Let m≥1, (r0=0,r1,…,rm) be a tuple of distinct real numbers and n≥2. We provide a characterization of those tuples (ω0,ω1,…,ωm) of real numbers such that there exist n-dimensional star bodies K,L with W˜rj(K,L)=ωj, j=0,…,m, where W˜r(K,L) denotes the r-th dual (relative) quermassintegral of K and L. This may be regarded as an analogue within the dual Brunn–Minkowski theory of Shephard's classification of quermassintegrals of two convex bodies.It turns out that the characterization of dual quermassintegrals is related to the moment problem, and based on this relation, we also derive new determinantal inequalities among the dual quermassintegrals. Moreover, this characterization will be the key tool in order to investigate structural properties of the set of roots of dual Steiner polynomials of star bodies.