The translative covering density 
of a convex body 
is the infimum of the asymptotic upper densities of coverings
of Euclidean space by translates
of 
.
Let 
have only finitely many points in each bounded
set, let ![Q_R=[-R,R]^n](/images/equations/TranslativeCoveringDensity/Inline5.svg)
, and require 
, where the sum is a Minkowski
sum. Then
 |
where 
denotes 
-dimensional
volume, 
is the number of centers
in the hypercube 
, and 
is the supremum limit.
The density counts overlaps with multiplicity.
The same infimum can be taken over periodic coverings with centers 
, where 
is a full-rank point lattice
and 
is a finite set of distinct coset representatives. Such a covering has density 
, where 
is the covolume.
The number of cosets is unrestricted. Requiring just one
coset gives the lattice
covering density 
, so 
.
Rogers (1957) proved the upper bound
 |
for every convex body 
with 
, where 
denotes the natural logarithm.
For instance, the hypercube ![[0,1]^n](/images/equations/TranslativeCoveringDensity/Inline22.svg)
has translative covering density 1.
