Local optimality of Zaks-Perles-Wills simplices

In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex S_{d,k} with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d \ge 3 and k \ge 1, the simplex S_{d,k} is a volume maximizer in the family P^d(k) of all d-dimensional lattice polytopes with k interior lattice points. To obtain a partial confirmation of this conjecture, one can try to verify it for a subfamily of P^d(k) that naturally contains S_{d,k} as one of the members. Currently, one does not even know whether S_{d,k} is optimal within the family S^d(k) of all d-dimensional lattice simplices with k interior lattice points. In view of this, it makes sense to look at even narrower families, for example, some subfamilies of S^d(k). The simplex S_{d,k} of Zaks, Perles and Wills has a facet with only one lattice point in the relative interior. We show that S_{d,k} is a volume maximizer in the family of simplices S \in S^d(k) that have a facet with one lattice point in its relative interior. We also show that, in the above family, the volume maximizer is unique up to unimodular transformations.