Mixed polytopes
In an ealier paper, \textit{W. Weil} [Geom. Dedicata 57, 91--103 (1995; Zbl 0838.52004)] has investigated the translative integral geometry of the centred support functional \(h^*\) (with respect to the Steiner point) of two convex bodies \(K\) and \(M\), and showed that \[ \int_{\mathbb{R}^n} h^*(K\cap (M+ x),\cdot)\,dx \] admits an expansion into \(n\) terms, of which the \(j\)th is homogeneous of degree \(j\) in \(K\) and \(n+ 1-j\) in \(M\). \textit{P. Goodey} and \textit{W. Weil} [Geom. Dedicata 99, 103--125 (2003; Zbl 1031.52002)] have further shown that this \(j\)th term is the support functional of a convex body, which is called a mixed body and is denoted \(T_j(K,M)\). The present author generalizes these ideas in the context of polytopes. He shows that the translation mixture of several polytopes similarly admits an expansion into terms which are homogeneous of various degrees in the component polytopes; these are the mixed polytopes of the title, and he obtains explicit formulae for their vertices and edges. A particular interesting feature is that he relates the construction to fibre polytopes of the Cartesian product of the components; see \textit{L. J. Billera} and \textit{B. Sturmfels} [Ann. Math. (2) 135, 527--549 (1992; Zbl 0762.52003)]. He also proves an inclusion inequality for translations mixtures, of which the extremal case characterizes simplices.
scientific article; zbMATH DE number 1969104