Prove that honeycomb structures are the most geometrically efficient structure

If you want to divide space up into uniform volume cells with minimum surface area, the honeycomb is not optimal. Look at the Weaire–Phelan structure. While honeycombs are not quite optimal, they are certainly close enough for bees -- they're suboptimal by only 0.3%.

It is called geometrically efficient because it is densely packed.