Gruber P. Convex and Discrete Geometry

a
f
b
o
e
2D
c
Fig. 30.3. Disc and triangular disc
In the second step of the proof, the following will be shown:
d
30 Packing of Convex Bodies 453
(9) Let D be an o-symmetric strictly convex disc. Then there is a triangular
disc C such that D = 1 2 (C − C) (Fig. 30.3).
To prove this, we first show that there is an affine regular convex hexagon H inscribed
in 2D. Fora ∈ bd 2D let H be the unique convex hexagon H inscribed in 2D with
opposite vertices a and −a such that its edges parallel to the line segment [a, −a]
both have length �a�. Since 2D is strictly convex and symmetric in o, these edges
are also symmetric in o. Thus H is affine regular.
Choose an affine regular hexagon with consecutive vertices a, b, c, d, e, f and
centre at o which is inscribed in 2D. Translate the triangles with vertices a, b, o; o, c,
d; f, o, e along with the adjacent lunae cut off from 2D by the line segments
[a, b], [c, d] and [ f, e], such that the triangles coincide. Let C be the union of the
translated triangles and lunae. Considering an o-symmetric convex hexagon circumscribed
to 2D the edges of which touch 2D at the points a,..., f, it follows that C
is triangular. Clearly, 2D = C − C, concluding the proof of (9).
In the third step of the proof, we show the following proposition:
(10) Let D be an o-symmetric strictly convex disc. Then the upper density of a
packing of translates of D never exceeds the density of the densest lattice
packing of D.
Let {D + t : t ∈ T } be a packing. By Proposition 30.2, its upper density is A(D)δ,
where δ is the upper density of T . Choose a triangular disc C such that D = 1 2 (C −
C). This is possible by (9). By Proposition 30.4, {C + t : t ∈ T } is also a packing.
Its upper density is A(C)δ. By (8), there is a packing lattice L of C and A(C)δ ≤
A(C)/d(L) by Corollary 30.1. Thus δ ≤ 1/d(L). Proposition 30.4 then shows that
L is also a packing lattice of D. Thus {D + l : l ∈ L} is a lattice packing of D
of density A(D)/d(L) ≥ A(D)δ. Now note that A(D)δ is the upper density of the
packing {D + t : t ∈ T }.
C