The Antipodal Defect of a Convex Polyhedron

Abstract

Problem C7 from the 2006 IMO Shortlist gives $A-B=V-1$ for a generic convex polyhedron $P\subset \mathbb R^3$, where $A$ counts antipodal vertex pairs and $B$ counts antipodal edge-midpoint pairs. We study arbitrary convex polyhedra through the defect $δ(P)=V(P)-1-A(P)+B(P)$. To $P$ we associate an 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 $H_2(X(P);\mathbb Z)\cong\mathbb Z^{δ(P)}$. In particular $δ(P)=β_2(X(P);\mathbb Q)\ge 0$, equivalently $A(P)-B(P)\le V(P)-1$. We also give an exact local formula for $δ(P)$ on the projective normal fan: it is the sum over exact opposite face pairs $\{F,G\}$ of $e(F)e(G)-(v(F)-1)(v(G)-1)$, equivalently in dimension three it is supported only on edge-facet and facet-facet exact pairs. This yields a facet-opposite formula, a zero-defect criterion, extremal bounds, and a spherical normal-graph profile. We further determine the integral lattice generated by square boundaries, obtaining the even-cycle lattice in the antipodal graph and Smith factors $1,\ldots,1,2$. Finally, we study the ordered representation space $\mathcal R(P)=\{(x,y)\in P\times P:x-y\in\partial(P-P)\}$ in all dimensions and show that it equivariantly deformation retracts onto $\partial(P-P)$, with unordered quotient homotopy equivalent to $\mathbb{RP}^{d-1}$.