MathDB

Consider a latticelike packing of translates of a convex region

K

. Let

t

be the area of the fundamental parallelogram of the lattice defining the packing, and let

t_{\min} (K)

denote the minimal value of

t

taken for all latticelike packings. Is there a natural number

N

such that for any

n>N

and for any

K

different from a parallelogram,

nt_{\min} (K)

is smaller that the area of any convex domain in which

n

translates to

K

can be placed without overlapping? (By a latticelike packing of

K

we mean a set of nonoverlapping translates of

K

obtained from

K

by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

Problem Statement