Nonnormal lattice polytopes at the counting scale

Construct, for every fixed dimension d at least 3, constants c and C greater than zero and, for every sufficiently large m, at least e^{cm} pairwise unimodularly inequivalent nonnormal d-dimensional lattice polytopes of normalized volume at most C m^{(d+1)/(d-1)}.

Background

Known enumeration results describe the number of unimodular-equivalence classes of fixed-dimensional lattice polytopes by volume scale. The problem asks whether nonnormal polytopes occur in exponentially many distinct classes at that same asymptotic counting scale.

References

For every fixed $d\geq3$, construct constants $c,C>0$ and, for every sufficiently large $m$, at least $e{cm}$ pairwise unimodularly not equivalent nonnormal $d$-dimensional lattice polytopes of normalized volume at most C m{(d+1)/(d-1)}.

— Most $(0,1)$-polytopes are not normal  (2609.02778 - Morales, 2 Sep 2026) in Problem Nonnormal polytopes at the counting scale, Section 5, Open problems