A spectral approach to polyhedral dimension

AbstractThespectrum spec() of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of. The number of non-zeros in spec() exceeds the dimension of by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of:(i)The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2q vertices (see Section 6).(ii)The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6).(iii)The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7).(iv)The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3q vertices (see Section 8). In the cases of (ii) and (iii), the associated dimension results are well-known. The dimension results for (i) and (iv) do not seem to be well-known.General principles are discussed for ‘balanced polytopes’ arising from complete structures.