Average-Sized and Normal-Sized Miniatures of Lattice Polytopes
A construction on a lattice polytope P that averages the volumes of its 'miniatures'—smaller similar copies contained in P with vertices on progressively finer lattices—to define an average-sized miniature (over all miniatures) and a normal-sized miniature (over only translated-and-rescaled copies). The main results give closed-form volume ratios: 2/15 of the area for any lattice square, and vol(P) over the binomial coefficient (2d+1 choose d) for any lattice simplex in d dimensions.
2501.00459
This paper studies 'miniatures' of a lattice polytope P: smaller copies M that are similar to P and contained in it, with vertices lying in the refined lattice (Z[1/n])^d. Averaging the volumes of al…