# A local maximizer for lattice width of $3$-dimensional hollow bodies

## TL;DR

This paper proves a local maximality result for a specific hollow tetrahedron's lattice width in 3D, showing it locally attains the maximum among nearby convex bodies, and constrains potential counterexamples.

## Contribution

The authors establish a local maximality property for a particular hollow tetrahedron's lattice width and provide bounds on possible counterexamples within tetrahedral space.

## Key findings

- The hollow tetrahedron of width 2+√2 is a local maximizer of lattice width.
- Potential counterexamples have width less than 3.972.
- Volume of potential counterexamples is within [2.653, 19.919].

## Abstract

The second and fourth authors have conjectured that a certain hollow tetrahedron $\Delta$ of width $2+\sqrt2$ attains the maximum lattice width among all three-dimensional convex bodies. We here prove a local version of this conjecture: there is a neighborhood $U$ of $\Delta$ in the Hausdorff distance such that every convex body in $U \setminus \{\Delta\}$ has width strictly smaller than $\Delta$. When the search space is restricted to tetrahedra, we compute an explicit such neighborhood. We also limit the space of possible counterexamples to the conjecture. We show, for example, that their width must be smaller than $3.972$ and their volume must lie in $[2.653, 19.919]$.