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.

  • $\begingroup$ You should probably read about packing and covering for lattices first; one source is Conway and Sloane, Sphere Packings, Lattices and Groups. $\endgroup$ – Will Jagy Jan 22 '15 at 5:59
  • $\begingroup$ @WillJagy : Do you think 'Fintie Packing and Covering' is too advanced to a starter in this area? $\endgroup$ – Nighty Jan 22 '15 at 6:09
  • $\begingroup$ Alright, your author, Boroczky, refers you to Appendix 13 for these notions, but that is just two or three pages. this is not an introductory book at all. You might try titles such as Geometry of Numbers for a more approachable introduction. The history is that Minkowski used lattices to solve various problems in the theory of numbers. $\endgroup$ – Will Jagy Jan 22 '15 at 6:09
  • $\begingroup$ this looks promising: maa.org/publications/ebooks/the-geometry-of-numbers $\endgroup$ – Will Jagy Jan 22 '15 at 6:15
  • $\begingroup$ In any case, read this: en.wikipedia.org/wiki/Geometry_of_numbers I do not really know the best approach for someone whose main interest is optimization; I do think you had better be pretty comfortable with lattices in a Euclidean space, and some of the simpler aspects in the wikipedia article. $\endgroup$ – Will Jagy Jan 22 '15 at 6:21

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 $K$.


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