Cone decompositions of non-simple polytopes (Q344360)

A polytope cone decomposition is presented, as follows.NEWLINENEWLINETheorem. Let \(P\subset V\) be any convex full-dimensional polytope in a vector space \(V\), and let \(S\) be the union of the affine spaces generated by the non-simple faces of \(P\). Then, for any \(\epsilon\in W\) and \(x\in V\setminus S\), we haveNEWLINENEWLINENEWLINE\[NEWLINE1_P^w(x) = \sum_{F\subseteq P} 1_P(\beta(\epsilon,\Delta_F))1_{T_F})^w(x),NEWLINE\]NEWLINENEWLINENEWLINEwhere \(\beta(\epsilon,\Delta_F)\) is the orthogonal projection of \(\epsilon\) onto the affine space \(\Delta_F\) generated by the face \(F\). Moreover, when all the weights assigned to the hyperplanes generated by the facets of \(P\) are equal to some fixed \(q\in\mathbb C\), the value of the function on the RHS at \(x\in S\) is also independent of the choice of regular triangulation \(T_F\) and is equal to \(q^{\mathrm{codim } F_x} - (r_x-\mathrm{codim } F_x)q^{\mathrm{codim }F_x-1}(1-q)\) if \(x\in P\) and \(0\) otherwise, where \(r_x\) is the number of facets of \(P\) that contain \(x\) and \(F_x\) is the face of \(P\) such that \(x\in \mathrm{int }F_x\). In particular, for \(q=1\), the RHS is equal to \(1_P(x)\), the unweighted characteristic function of \(P\), for every \(x\in V\).NEWLINENEWLINEA few remarks: \(W\subset V\) is the set of vectors \(\epsilon\) such that \(\beta(\epsilon, \Delta_F)\) is not in any affine space generated by a proper subface of \(F\), \(w\) stands for \textit{weight}, i.e. each facet \(F\) of \(P\) is provided with a weight \(q_F\) and the points in the intersection of facets are provided with the product of corresponding weights, so \(1^w\) is a weighted characteristic function. The vector \(\epsilon\) is responsible for \textit{polarization}: in several polytope cone decompositions we \textit{flip} vectors in cones, according to \(\epsilon\), cf. Lawrence-Varchenko decomposition.NEWLINENEWLINEThe proof goes as follows: such a formula is known for simple polytopes [the authors, Adv. Math. 214, No. 1, 379--416 (2007; Zbl 1134.52013)], where the motivation was to study the critical points of the square of the module on the moment map, and a particular application was a new formula for the sum of the values of a function over the lattice points in a polytope (Euler-Maclaurin formula). At a non-simple face \(F\) of \(P\), \textit{the regular triangulations} of the cone dual to the normal cone of \(F\) parametrize all combinatorial types of small perturbation of \(P\) near \(F\) by parallel shifts of facets, which make the polytope simple near \(F\). Therefore each of regular triangulations \(T_F\) provide a way to present the tangent cone to \(P\) at \(F\) as an intersection of simple cones, which are then \textit{polarized}. Later it is proved that a choice of a regular triangulation is irrelevant.NEWLINENEWLINEMost known cone polytope decomposition formula presents \(1_P\) as the sum of tangent cones to all faces [\textit{C. Brianchon}, ``Théorème nouveau sur les polyèdres'', J. École NEWLINEPolytechnique 15, 317--319 (1837)], another formula presents \(1_P\) as the sum of polarized tangent cones at the vertices of \(P\), the Lawrence-Varchenko decomposition [\textit{J. Lawrence}, Math. Comput. 57, No. 195, 259--271 (1991; Zbl 0734.52009); \textit{A. N. Varchenko}, Funkts. Anal. Prilozh. 21, No. 1, 11--22 (1987; Zbl 0615.52005)].