Improving bounds on the diameter of a polyhedron in high dimensions

In 1992, Kalai and Kleitman proved that the diameter of a d-dimensional polyhedron with n facets is at most n2+log2d. In 2014, Todd improved the Kalai-Kleitman bound to (n−d)log2d. We improve the Todd bound to (n−d)−1+log2d for n≥d≥7, (n−d)−2+log2d for n≥d≥37, and (n−d)−3+log2d+O1∕d for n≥d≥1.