# Hollow polytopes of large width

## TL;DR

This paper constructs the first known high-width hollow lattice polytopes and simplices, demonstrating that maximum lattice width grows at least linearly with dimension, and explores bounds in three dimensions.

## Contribution

It introduces the first known hollow lattice polytopes of width larger than their dimension and establishes linear growth of maximum lattice width with dimension.

## Key findings

- Constructed hollow lattice polytopes of dimension 14 and width 15.
- Constructed hollow lattice simplices of dimension 404 and width 408.
- Showed that maximum lattice width grows at least linearly with dimension.

## Abstract

We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a hollow (non-lattice) tetrahedron of width $2+\sqrt2$ and conjecture that this is the maximum width among $3$-dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with $d$. In particular, the constructions above imply the existence of hollow lattice polytopes (resp. hollow simplices) of arbitrarily large dimension $d$ and width $\simeq 1.14 d$ (resp.$~\simeq 1.01 d$).