Geometriae Dedicata

, Volume 13, Issue 2, pp 149–155

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

• W. Kuperberg
Article

DOI: 10.1007/BF00147658

Kuperberg, W. Geom Dedicata (1982) 13: 149. doi:10.1007/BF00147658

## 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.