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 $\delta (P)=V(P)-1-A(P)+B(P)$. We construct a two-dimensional antipodal square complex $X(P)$ and prove $H_0(X(P);\mathbb{Z})\cong \mathbb{Z}$, $H_1(X(P);\mathbb{Z})\cong \mathbb{Z}\mathrm{/}2$, and $H_2(X(P);\mathbb{Z})\cong {\mathbb{Z}}^{\delta (P)}$. Consequently, $\delta (P)\ge 0$, extending the generic identity to the inequality $A(P)-B(P)\le V(P)-1$. 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 $1,\dots ,1,2$. Applications include a zero-defect criterion, centrally symmetric and extremal formulas, and an explicit description of defects and primitive belts for pyramids over polygons.