, Volume 13, Issue 2, pp 149-155

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

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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.