TOPICS
Search

Lattice Covering Density

The lattice covering density theta_L(K) of a convex body K subset R^n is the infimum of the densities of coverings of Euclidean space by translates of K whose centers form one full-rank point lattice (Rogers 1957). If Lambda=BZ^n, where B is an n×n invertible matrix, then its covolume is covol(Lambda)=|detB|, and

theta_L(K)=inf_(Lambda:K+Lambda=R^n)(vol_n(K))/(covol(Lambda)),

where vol_n denotes n-dimensional volume and K+Lambda is the Minkowski sum. The ratio counts overlaps with multiplicity and is the average number of covering translates over a fundamental parallelepiped.

Every lattice covering is also a translative covering, so 1<=theta_T(K)<=theta_L(K), where theta_T is the translative covering density. For the hypercube K=[0,1]^n, the point lattice Z^n gives theta_T(K)=theta_L(K)=1.

See also

Convex Body, Covolume, Fundamental Parallelepiped, Lattice Covering Radius, Octahedron Covering, Point Lattice, Translative Covering Density

Explore with Wolfram|Alpha

References

Rogers, C. A. "A Note on Coverings." Mathematika 4, 1-6, 1957. https://doi.org/10.1112/S0025579300001030.

Cite this as:

Weisstein, Eric W. "Lattice Covering Density." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LatticeCoveringDensity.html

Subject classifications