Packing Problem - What do these notations mean?

Question

I am reading 'Finite Packing and Covering' and I find some notations on the first few pages that are not defined in the book. I am guessing those are standard in the discussion of packing problems. As I cannot find similar discussion on the Internet, I would like to see if anyone can clear my confusion.

Let $K$ be a convex domain. ( I guess it means a convex body? ) Given an arrangement of congruent copies of $K$ that is periodic with respect to some lattice $\Lambda$ and given $m$ equivalence classes (what equivalence classes are we talking about? ), it is natural to call $m \cdot A(K) / \det \Lambda$ the density of the arrangement. (What is $A(K)$? )

... We define $\Delta(K)=A(K)/\delta(K)$, where $\delta(K)$ is the packing density. (What is the meaning of this $\Delta(K)$?)

Any help would be appreciated.

Answer 1

What I suppose from just readng your quote:

• A domain is an open connected set; as convex implies connected, $K$ is just a convex open set.
• I suppsoe that $A(K)$ is the area (or volume or most general Lebesgue measuer) of $K$
• It is said that the packing is periodic with lattice $\Lambda$, but that does not mean that all those copies of $K$ are centered only at the lattice points (in fact, if some are rotated nontrivially, this would contradict the periodicity claim); also nobody forbds that you consider only a sublattice of $\Lambda$. At any rate, the action of $\Lambda$ on the set of copies of $K$ induces an equivalence relation on these copies. In other words, there are $m$ essentially different copies of $K$ and then all other copies are obtained by translatins accoring to $\Lambda$. In other paralance, $m$ is the number of copies of $K$ in an "elementary cell". Indeed, since the volume of such an elementary cell is precisely $\det(\Lambda)$, the suggestion to call $mA(K)/\det(\Lambda) the density of the packing should become eveident. As one might expect, this expression also coincides with the limite relative proportion of a large ball that is occupied by the packing.

From $\Delta(K)=A(K)/\delta(K)=\det(\Lambda)/m$ (using the above definition of density), we read that $\Delta(K)$ denotes the average space that "belongs" to each copy of $K$ (i.e., incluing a fair share of the "gaps"). I am only surprised abot the notatin $\delta(K)$ (and $\Delta(K)$; after all, these depend on the actual packing and not just on our packee $K$.