Reinhardt's problem of lattice packings of convex domains: Local extremality of the Reinhardt octagon

Abstract

I. Reinhardt [6] considered the problem of finding a convex domain n centrally symmetric about the origin ~ such that the density 6{n) of the densest lattice packing of ~ is minimal. This is equivalent to finding a domain n of minimal area VC~) with critical determinant,As i (for the basic concepts of the geometry of numbers, see [3]). Reinhardt conjectured that such a domain is a ~0-regular octagon smoothed by hyperbolic arcs (for construction see [6]). Reinhardt's problem has been discussed in [6, 4, 5, 2, i]. We will show that ~) has a strict local minimum on •0More precisely, we will prove