Geometriae Dedicata

, Volume 13, Issue 2, pp 149–155 | Cite as

Packing convex bodies in the plane with density greater than 3/4

  • W. Kuperberg
Article

Conclusion

By an argument of a topological nature, the Theorem whose proof has just been completed can be somewhat strengthened. Since the collection of affine equivalence classes of all convex plane bodies of area 1 is a compact set, and since the function assigning to each such equivalence class the minimum area of a p-hexagon containing a representative of that class is continuou s,there exists a minimum value for that function, taken on a specific element of that compact set. Let us denote that minimum value by Δ. We have proved in this \(\Delta < \frac{4}{3}\), thus we can conclude that there exists a number \(d > \frac{3}{4}\) (namely \(d = \Delta ^{ - 1} \)) such that every convex body can be packed in the plane with density at least d. The value of A remains unknown.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chakerian, G. D. and Lange, L. H., ‘Geometric Extremum Problems’. Math. Mag. 44 (1971), 57–69.Google Scholar
  2. 2.
    Courant, R., ‘The Least Dense Lattice Packing of Two-Dimensional Convex Bodies’. Comm. Pure Appl. Math. 18 (1965), 339–343.Google Scholar
  3. 3.
    Fáry, I., ‘Sur la densité des reseaux de domaines convexes’. Bull. Soc. Math. France 78 (1950), 152–161.Google Scholar
  4. 4.
    Fejes Tóth, L., Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953.Google Scholar
  5. 5.
    Kuperberg, W., ‘On Minimum Area Quadrilaterals and Triangles Circumscribed about Convex Plane Regions’. El. Math., to appear.Google Scholar

Copyright information

© D. Reidel Publishing Company 1982

Authors and Affiliations

  • W. Kuperberg
    • 1
  1. 1.Department of MathematicsAuburn UniversityAuburn UniversityUSA

Personalised recommendations