Math 2.0

Convex and discrete geometry Definition

Covering density

Builds on Lebesgue measureLimit inferiorClosed ballPacking density

Definition

Let n≥1n \ge 1, let vol⁡\operatorname{vol} be the Lebesgue measure on ℝn\R^n, and let BRB_R be the closed ball of radius RR centred at 00. A covering of ℝn\R^n is a countable family 𝒞=(Ki)i∈I\mathcal C = (K_i)_{i \in I} of compact sets whose union is ℝn\R^n. It is required that each bounded set meets only finitely many KiK_i. The lower density of 𝒞\mathcal C is d‾(𝒞)=lim inf⁡R→∞∑i∈Ivol⁡(Ki∩BR)vol⁡(BR),\underline{d}(\mathcal C) = \liminf_{R \to \infty} \frac{\sum_{i \in I} \operatorname{vol}(K_i \cap B_R)}{\operatorname{vol}(B_R)}, with the limit inferior; when the limit exists it is the density of 𝒞\mathcal C. Let KK be a compact set with nonempty interior. The covering density of KK is ϑ(K)=inf⁡d‾(𝒞)\vartheta(K) = \inf \underline{d}(\mathcal C), over all coverings 𝒞\mathcal C of ℝn\R^n by congruent copies of KK.

Remarks

The ratio is the same as for the packing density; packings forbid overlaps, coverings forbid gaps. Since the sets cover BRB_R, the ratio is at least 11, so ϑ(K)≥1\vartheta(K) \ge 1, with equality when KK tiles space, as a cube does. Example: unit discs centred at the points of the triangular lattice spanned by (3,0)(\sqrt 3, 0) and (3/2,3/2)(\sqrt 3/2, 3/2). Each disc contains the regular hexagon of circumradius 11 around its centre, and these hexagons tile the plane, so the discs cover. The density is π\pi divided by the hexagon area 33/23\sqrt 3/2, that is 2π/272\pi/\sqrt{27}. Non-example: unit discs centred at 2ℤ22\Z^2 do not form a covering, since the point (1,1)(1, 1) is at distance 2>1\sqrt 2 > 1 from every centre. For a lattice covering K+ΛK + \Lambda, the density exists and equals vol⁡(K)/det⁡Λ\operatorname{vol}(K)/\det \Lambda.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptClosed ball, concept, no evidence levelClosed ballconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptPacking density, concept, no evidence levelPacking densityconceptSequence, definition, no evidence levelSequencedefinitionLebesgue measure, object, no evidence levelLebesgue measureobjectReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit inferior, definition, no evidence levelLimit inferiordefinitionCovering density, definition, no evidence levelCovering densitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Closed ball concept
  3. Euclidean space object
  4. Outer measure concept
  5. Packing density concept
  6. Sequence definition
  7. Lebesgue measure object
  8. Real sequence definition
  9. Bounded sequence definition
  10. Limit inferior definition
  11. Covering density definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Covering_problems
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Kinddefinition
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MSC52C17, 52C15, 11H31
DefinesCovering density
UsesLebesgue measure, Limit inferior, Closed ball, Packing density

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