Reinhardt Conjecture (Smoothed Octagon Minimizes Packing Density)

Determine whether the smoothed octagon uniquely achieves the least greatest packing density among all centrally symmetric convex disks in the plane, up to affine transformation, and establish that this density equals (8 − √32 − ln 2)/(√8 − 1) ≈ 0.902414. This is the full Reinhardt conjecture asserting the smoothed octagon is the most unpackable centrally symmetric convex disk.

Background

The Reinhardt problem asks for the centrally symmetric convex disk in the plane with the least greatest packing density. In 1934, Karl Reinhardt conjectured that a specific shape—the smoothed octagon—minimizes this density, and gave an explicit formula for its value. The book proves Mahler’s First conjecture (that the most unpackable disk is a smoothed polygon) but explicitly notes that the full Reinhardt conjecture remains unresolved.

Multiple places in the text emphasize the open status of the conjecture: the Abstract directly states that Mahler’s Second conjecture (identical to Reinhardt’s conjecture) remains open, and the Introduction reiterates that the conjecture has remained open since 1934.