Oda's conjecture asks whether every smooth lattice polytope has the integer decomposition property: can every lattice point in any dilation kP be written as a sum of k lattice points in P? The conjecture connects the geometric smoothness of a polytope — every vertex cone is generated by a lattice basis — to an arithmetic property about decomposing lattice points.
Hegedüs proves Oda's conjecture for smooth Fano polytopes. A Fano polytope contains the origin in its interior and has all vertices as primitive lattice points; smoothness further requires that the vertex cones are unimodular. The proof establishes that the delta-vector — a combinatorial invariant encoding the Ehrhart polynomial's arithmetic — is unimodal, and derives volume bounds for these polytopes.
The integer decomposition property (IDP) is not automatic even for very structured polytopes. A lattice polytope can be convex, bounded, full-dimensional, with all vertices on the lattice, and still fail IDP. The failure mode is subtle: a lattice point in 2P might not decompose as a sum of two lattice points in P because the midpoint of the connecting line segment misses the lattice. Smoothness eliminates this failure by guaranteeing that the local geometry at every vertex is lattice-equivalent to the standard simplex, where decomposition is trivial.
The conjecture remains open for general smooth lattice polytopes. The Fano case works because the origin-interior condition provides global control — every lattice point in the polytope can be “seen” from the origin through unimodular vertex cones. Without this central anchor, the local-to-global passage from vertex smoothness to global decomposability remains unresolved.