On the Reinhardt conjecture (Q763577)

The article under review suggests a detailed strategy of a proof of the Reinhardt Conjecture. Let \(K\) be a convex centrally symmetric domain in~\({\mathbb R}^2\). Given a rank 2 lattice \(\Lambda \subset {\mathbb R}^2\) such that the translates of \(K\) by points of \(\Lambda\) have disjoint interiors, define the packing density of \(\Lambda+K\) to be \[ \delta_{\Lambda}(K) = \frac{\text{Area}(K)}{\det(\Lambda)}, \] and let \(\delta(K)\) be the maximum of \(\delta_{\Lambda}(K)\) taken over all such lattice packings of~\(K\). Let \[ \delta_{\min} = \inf \{ \delta(K) : K \text{ a convex centrally symmetric domain in } {\mathbb R}^2 \}. \] Reinhardt proved that \(\delta_{\min}\) is attained, and posed the problem to determine \(\delta_{\min}\) and to explicitly describe \(K\) for which \(\delta(K)=\delta_{\min}\). Furthermore, he conjectured that this \(K\) is a smoothed octagon, that is a regular octagon the corners of which are clipped with hyperbolic arcs chosen so that there is a unique tangent at each point of the boundary. It is known that for such an octagon \(K\), \[ \delta(K) = \frac{8-\sqrt{32}-\ln 2}{\sqrt{8}-1} \approx 0.902414. \] The author of the present paper gives a brief overview of previous work done in the direction of this conjecture. He then outlines an approach to the conjecture, using calculus of variations and nonlinear optimization techniques.