Gruber P. Convex and Discrete Geometry

Lattice Packing and Packing of Translates in E 2
We show the following result of Fejes Tóth and Rogers:
30 Packing of Convex Bodies 451
Theorem 30.5. Let C be a proper convex disc in E 2 . Then δT (C) = δL(C).
The following proof was given by Fejes Tóth [331]. It seems to be the shortest
proof known.
Proof. Since the strictly convex discs are dense among all convex discs, it is sufficient
to prove this result for strictly convex discs. The result is first proved for
triangular discs, that is strictly convex discs C contained in a convex hexagon with
vertices a, b, c, d, e, f, parallel opposite edges and such that a, c, e ∈ C. Callthe
line through a and b vertical, a and e the left and right and c the bottom vertex of C
(Fig. 30.1).
The first step in the proof of the theorem is to show the following:
(1) Let {C + t : t ∈ T } be a packing of translates of a triangular disc C. Then
for each translate C + t there is an associated set At such that:
(i) diam At ≤ 3diamC.
(ii) The sets At ∪ (C + t), t ∈ T, do not overlap.
(iii) The density of C + t in At ∪ (C + t), i.e. the quotient of the areas
of these sets, equals the density of a certain lattice packing of C.
Given C + t, consider translates C + r, C + s, not necessarily belonging to the
given packing, such that the bottom vertices of C + s and C + r coincide with the
left and right vertices of C + t. The translates C +r, C + s, C + t enclose a region A.
We distinguish two cases (Fig. 30.2):
First, no translate C + u, u ∈ T, overlaps A. Then put At = A. The region At is
contained in a triangle, a congruent copy of which is contained in C + t. Hence
Further,
(2) diam At ≤ diam C.
(3) each point of At is connected to a point of C + t by a vertical line segment
and this line segment does not meet any translate C + u, u ∈ T \{t}.
a
b
f
C
c
Fig. 30.1. Triangular disc
e
d