Combinatorial coefficients and the mixed volume of polytopes
Let \(\Delta= \Delta_1+ \cdots +\Delta_n\) be the Minkowski sum of an ordered \(n\)-tuple of convex polytopes \(\Delta_i\) in \(\mathbb{R}^n\). The \(n\)-tuple is said to be expanded if for each face \(\Gamma= \Gamma_1 +\cdots +\Gamma_n\) of \(\Delta\) among its summands \(\Gamma_i \subset \Delta_i\) there is at least one vertex. The \(n\)-tuple is said to be inductively expanded if, in addition for any \(m(0 <m <n)\), the set of the first \(m\) summands of an \(m\)-dimensional face \(\Gamma^m =\Gamma_1^m +\cdots +\Gamma^m_n\) of \(\Delta\) is expanded whenever the last \(n-m\) summands of \(\Gamma^m\) are vertices. For such special polytopes \(\Delta\) the author gives an explicit formula for the combinatorial coefficients \(k_v\) of vertices \(v\) of \(\Delta\) which means the local degree of a germ of a special continuous mapping of the boundary of \(\Delta\) upon the boundary of the positive cone (Theorem 1). Further he proves the following formula for the mixed volume of inductively expanded polytopes \[ n!V(\Delta_1, \dots, \Delta_n) =(-1)^n \sum k_v \text{det} (v_1, \dots, v_n) \] (theorem 2) conjectured earlier by the author and A. G. Khovanskii. The summation extends over all vertices \(v\in \Delta\), and \(v_i\) are the vertices of \(\Delta_i\) such that \(v=v_1 +\cdots +v_n\).
scientific article; zbMATH DE number 1023826