The Antipodal Defect of a Convex Polyhedron
Problem C7 from the 2006 IMO Shortlist gives \(A(P)-B(P)=V(P)-1\) for a generic convex polyhedron, where \(A(P)\) counts antipodal vertex pairs and \(B(P)\) counts antipodal edge-midpoint pairs. For an arbitrary convex polyhedron \(P\subset\mathbb{R}^3\), define \(δ(P)=V(P)-1-A(P)+B(P)\). We construct a two-dimensional antipodal square complex \(X(P)\) and prove \(H_0(X(P);\mathbb{Z})\cong\mathbb{Z}\), \(H_1(X(P);\mathbb{Z})\cong\mathbb{Z}/2\), and \(H_2(X(P);\mathbb{Z})\cong\mathbb{Z}^{δ(P)}\). Consequently, \(δ(P)\geq 0\), extending the generic identity to the inequality \(A(P)-B(P)\leq V(P)-1\). The proof uses a directed double cover, a polyhedral support blow-up over the normal sphere, and the Vietoris--Begle mapping theorem. Independently, Euler integration on the projective normal fan gives an exact local formula for the defect; in three dimensions, only exact edge--facet and facet--facet opposite pairs contribute. We also determine the integral image of the square-boundary map: it is the even-cycle lattice of the antipodal graph, with nonzero Smith invariant factors \(1,\ldots,1,2\). Applications include a zero-defect criterion, centrally symmetric and extremal formulas, and an explicit description of defects and primitive belts for pyramids over polygons.