338 VIII The Geometry of Numbers
We can chooseε>0 so small that the ballsBε+x 1 andBε+x 2 are disjoint and
contained inG.ThenG′=G(Bε+x 1 )is a bounded nonempty open set with closure
K′=K(intBε+x 1 ).Since
Bε+x 1 =Bε+x 2 +a⊆K′+a,theΛ-translates ofK′containKand therefore also coverRn. Hence, by what we have
already proved,λ(K′)≥d(Λ).Sinceλ(K)>λ(K′), it follows thatλ(K)>d(Λ).
Suppose now that theΛ-translates ofKare a packing ofRn.ThenΛdoes not
contain the difference of two distinct points in the interiorGofK,sinceG+aand
G+bare disjoint ifa,bare distinct points ofΛ. It follows from Proposition 9 that
λ(K)=λ(G)≤d(Λ).Suppose, in addition, that theΛ-translates ofKdo not coverRn. Thus there exists
a pointy∈Rnwhich is not in anyΛ-translate ofK. We will show that we can choose
ε>0sosmallthatyis not in anyΛ-translate ofK+Bε.
If this is not the case then, for any positive integerv, there existsav∈Λsuch that
y∈K+B 1 /v+av.Evidently the sequenceavis bounded and hence there existsa∈Λsuch thatav=a
for infinitely manyv.Buttheny∈K+a, which is contrary to hypothesis.
We may in addition assumeεchosen so small that|x|> 2 εfor every nonzero
x∈Λ. Then the setS=G∪(Bε+y)has the property thatΛdoes not contain the
difference of any two distinct points ofS. Hence, by Proposition 9,λ(S)≤d(Λ).Since
λ(K)=λ(G)<λ(S),it follows thatλ(K)<d(Λ).
We next apply Proposition 9 to convex sets. Minkowski’s lattice point theorem
(Theorem 1) is the special casem=1(andΛ=Zn) of the following generalization,
due to van der Corput (1936):
Proposition 12Let C be a symmetric convex subset ofRn,Λa lattice inRnwith
determinantd(Λ), and m a positive integer.
Ifλ(C)> 2 nmd(Λ), or if C is compact andλ(C)= 2 nmd(Λ), then there exist
2 m distinct nonzero points±y 1 ,...,±ymofΛsuch that
yj∈C ( 1 ≤j≤m),
yj−yk∈C ( 1 ≤j,k≤m).Proof The setS={x/2:x∈C}has measureλ(S)=λ(C)/ 2 n. Hence, by Proposi-
tion 9, there existm+1 distinct pointsx 1 ,...,xm+ 1 ∈Csuch that(xj−xk)/ 2 ∈Λ
for allj,k.
The vectors ofRnmay be totally ordered by writingx>x′ifx−x′has its first
nonzero coordinate positive. We assume the pointsx 1 ,...,xm+ 1 ∈Cnumbered so that
x 1 >x 2 >···>xm+ 1.