Empty simplices of large width

An empty simplex is a lattice simplex in which vertices are the only lattice points. After reviewing what is known about width of lattice polytopes and convex bodies, we show two constructions leading to the first known empty simplices of width larger than their dimension:

(1) We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we find five empty ones of width $11$, and none of larger width.
(2) Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and of width growing asymptotically as $d/$arcsinh$(1) \sim 1.1346\,d$.

The width in part (2) is (asymptotically) only $3\%$ lower than the widest convex bodies known.

This is joint work with Joseph Doolittle, Lukas Katthän, and Benjamin Nill.