# On the Reinhardt Conjecture

@article{Hales2011OnTR, title={On the Reinhardt Conjecture}, author={Thomas C. Hales}, journal={arXiv: Metric Geometry}, year={2011} }

In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density. He conjectured that the unique solution up to an affine transformation is the smoothed octagon (an octagon rounded at corners by arcs of hyperbolas). This article offers a detailed strategy of proof. In particular, we show that the problem is an instance of the classical problem of Bolza in the calculus of variations. A minimizing solution is known to exist. The…

## 5 Citations

The Reinhardt Conjecture as an Optimal Control Problem

- Mathematics
- 2017

In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is…

On the smallest area $n-1$-gon containing a convex $n$-gon

- Mathematics
- 2021

where P varies in the set Pm of all convex m-gons, and, for a fixed m-gon P , the minimum is taken over all n-gons Q containing P ; here | · | denotes area. It has been proved that r(3, 4) = 2, and…

Mathematics in the age of the Turing machine

- Computer ScienceTuring's Legacy
- 2014

The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs, as well as some examples from the literature.

Errata and Addenda to Mathematical Constants

- Mathematics
- 2016

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always…

## References

SHOWING 1-10 OF 15 REFERENCES

The colossal book of mathematics : classic puzzles, paradoxes, and problems : number theory, algebra, geometry, probability, topology, game theory, infinity, and other topics of recreational mathematics

- Art
- 2001

Whether discussing hexaflexagons or number theory, Klein bottles or the essence of "nothing," Martin Gardner has single-handedly created the field of "recreational mathematics." The Colossal Book of…

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

- Mathematics
- 1988

Local extremality of the Reinhardt octagon is proved.

Calculus of Variations and Optimal Control Theory

- Mathematics
- 1967

M. R. Hestenes London: John Wiley. 1967. Pp. xii + 405. Price £5. Professor Hestenes has been a leading researcher on optimization theory since the early 1930's and this book contains much of his…

Combinatorial geometry

- MathematicsWiley-Interscience series in discrete mathematics and optimization
- 1995

A bound on the critical determinant of a two-dimensional convex symmetric domain

- Izv. Vyssh. Uchebn. Zaved. Mat., 103:103–107
- 1970