Gruber P. Convex and Discrete Geometry

474 Geometry of Numbers
Remark. Groemer [400] proved that a convex polytope which admits a tiling of E d
by (positive) homothetic copies, is already a translative tile and thus a parallelohedron
by Theorems 32.2 and 32.3.
Sufficiency of the Conditions
More difficult is the proof of the following converse of Theorem 32.2:
Theorem 32.3. Let P be a proper convex body in E d which satisfies Properties (i)–
(iv) in Theorem 32.2. Then P is a translative tile, more precisely even a lattice tile;
i.e. it is a parallelohedron.
Proof. In the following int is the interior relative to E d or a sphere S and by relint
we mean the interior relative to the affine or the spherical hull. We may assume that
o is the centre of P.
In the first step we describe a family of translates of P which is a candidate for a
facet-to-facet lattice tiling of P. IfF is a facet of P, then the facet −F is a translate
of F (note (ii) and (iii) and that o is the centre of P), say F =−F + tF. Clearly,
P ∩ (P + tF) = F =−F + tF. Now consider the following family of translates
of P:
(4) {P + l : l ∈ L}, where L = � � u FtF : F facet of P, u F ∈ Z � .
We will show that this is the desired facet-to-facet lattice tiling of P. The following
notation will be needed. Given l ∈ L and a face F of the translate P + l, letL(l, F)
be the subset of L defined recursively as follows: l ∈ L(l, F) and n ∈ L(l, F) if
there is a point m ∈ L(l, F) such that P + m and P + n have a facet in common
which contains F as a face. Clearly, L(o, ∅) = L and L(o, P) ={o}. Since L is an
additive sub-group of E d and thus L(l, F) = L(o, F − l) + l, it is sufficient to study
L(o, F) where F is a face of P. Since P contains only finitely many faces which are
translates of F,
(5) L(l, F) is finite for each l ∈ L and each face F �= ∅of P + l.
In the second step we will show that
(6) {P + l : l ∈ L} is a covering of E d .
To see this, it will be proved first that
(7) for k = d, d − 1,...,0, one has the following inclusion:
relint F ⊆ int � � P + l : l ∈ L(o, F) � for each face F of P with
dim F = k.
Obviously, (7) holds for k = d (where L(o, P) = {o}) and k = d − 1 (where
L(o, F) ={o, tF}). Assume now that k < d − 1 and (7) holds for all faces F of P
with dim F = k + 1. Let G be a face of P with dim G = k. LetS be a sphere of
dimension d − k − 1 centred at a point of relint G, orthogonal to aff G, and so small
that it meets only those faces of P + l, l ∈ L(o, G), which contain G. For the proof
that (7) holds for the face G, it is sufficient to show that
(8) S ⊆ � � P + l : l ∈ L(o, G) � .