Reinhardt's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A centrally symmetric convex body $K\subset\mathbb R^2$ is compact, convex, has nonempty interior, and is invariant under reflection in some point. A full-rank lattice $\Lambda$ gives a packing if the translates $K+v$, $v\in\Lambda$, have disjoint interiors. Its density is $|K|/\operatorname{covol}(\Lambda)$, where the covolume is the area of a fundamental parallelogram. Let $\delta(K)$ be the maximum of these lattice-packing densities. For this class of planar bodies it also equals the best unrestricted packing density. Reinhardt's smoothed octagon is obtained by clipping the corners of a regular octagon by hyperbolic arcs, meeting the remaining sides with continuous tangents; each arc's asymptotes extend two sides of the original octagon. Its density is \[c_R=\frac{8-\sqrt{32}-\log 2}{\sqrt8-1}.\] The conjecture includes uniqueness up to invertible affine transformations.

Statement recorded in the supplied source.

Does every centrally symmetric convex planar body satisfy $\delta(K)\geq c_R$, with equality precisely for affine images of Reinhardt's smoothed octagon? [S2]

Short English statement

Is Reinhardt’s octagon with hyperbolically rounded corners the unique shape, up to affine changes, whose densest packing has the smallest density among centrally symmetric convex planar bodies?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200029 (AMR-071-0029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Thomas C. Hales, On the Reinhardt conjecture (2011). Abstract and Section 1, pp. 1–2. https://arxiv.org/pdf/1103.4518.