An isoperimetric problem for polyhedra

An old theorem states that among all polygons of given area that tile the plane by translation, the regular hexagon has least perimeter.  This talk will discuss the solution of the corresponding problem in three dimensions:  among all convex polyhedra of given volume that tile space by translation, the truncated octahedron of Archimedes has least surface area.  This result affirms a conjecture made by K. Bezdek in 2006.
This research is joint work with Lark Song, in partnership with AI.  Part of the talk will be a discussion of my experience with AI as a research tool. Reference: arXiv:2609.00997