25,351 research outputs found
Lack of Sphere Packing of Graphs via Non-Linear Potential Theory
It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1}
or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z,
in R^d, for all d. A similar result is proved for some other graphs too. Rather
than using a direct geometrical approach, the main tools we are using are from
non-linear potential theory.Comment: 10 page
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
Decomposition of multiple packings with subquadratic union complexity
Suppose is a positive integer and is a -fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most sets. Suppose there is a function
with the property that any members of determine
at most holes, which means that the complement of their union has at
most bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that can be
decomposed into at most (-fold) packings, where is a constant
depending only on and .Comment: Small generalization of the main result, improvements in the proofs,
minor correction
On the equality of Hausdorff measure and Hausdorff content
We are interested in situations where the Hausdorff measure and Hausdorff
content of a set are equal in the critical dimension. Our main result shows
that this equality holds for any subset of a self-similar set corresponding to
a nontrivial cylinder of an irreducible subshift of finite type, and thus also
for any self-similar or graph-directed self-similar set, regardless of
separation conditions. The main tool in the proof is an exhaustion lemma for
Hausdorff measure based on the Vitali Covering Theorem.
We also give several examples showing that one cannot hope for the equality
to hold in general if one moves in a number of the natural directions away from
`self-similar'. For example, it fails in general for self-conformal sets,
self-affine sets and Julia sets. We also give applications of our results
concerning Ahlfors regularity. Finally we consider an analogous version of the
problem for packing measure. In this case we need the strong separation
condition and can only prove that the packing measure and -approximate
packing pre-measure coincide for sufficiently small .Comment: 21 pages. This version includes applications concerning the weak
separation property and Ahlfors regularity. To appear in Journal of Fractal
Geometr
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