Mixed volumes for nonconvex bodies (Q1896885)

Two not necessary convex bodies \(K\), \(K' \subseteq \mathbb{R}^n\) have the same behavior with respect to mixed volumes if for all convex bodies \(K_1, \dots, K_n\) holds \(V(K, K_2, \dots, K_n) = V(K', K_2, \dots, K_n)\). The definition of the mixed volume \(V(K_1, \dots, K_n)\) for nonconvex bodies goes back to \textit{H. Groemer} [Geom. Dedicata 6, No. 2, 141-163 (1977; Zbl 0394.52003)]. The author gives necessary and sufficient conditions for two bodies to have the same behavior in the case of nonconvex polyhedra (Theorem 4.4) and in the case of (nonconvex) smooth bodies (Theorem 7.2). Clearly, two convex bodies with the same behavior are equal up to a translation.