Oda's conjecture for reflexive polytopes: some special cases

In this talk, we show that Oda's question (the IDP pair property) holds for an $n$-dimensional simplicial reflexive polytope $P$ and a lattice polytope $Q$ containing the origin when the face fan of $P$ refines that of $Q$, provided that $P$ has at most $n+1$ lattice points on each facet and admits a unimodular triangulation. We then observe that the statement also holds for any two facet-unimodular polytopes all of whose facet normals have at most two non-zero entries, e.g. dual of symmetric edge polytopes. In general, two facet-unimodular polytopes do not necessarily become an IDP pair, but if we additionally require two reflexive polytopes to be co-unimodular, then they form an IDP pair. Finally, we present a generalization of this result.