Math 2.0

Convex and discrete geometry Discrete and combinatorial geometry Problem

Danzer set

Builds on Convex bodyLebesgue measure

Problem

Let d≥2d \ge 2. A set D⊆ℝdD \subseteq \R^d is a Danzer set if it meets every convex body in ℝd\R^d of volume 11, where volume is Lebesgue measure. Is there a Danzer set DD of bounded density? This means that there is a constant CC such that every ball of radius r≥1r \ge 1 contains at most CrdC r^d points of DD.

Remarks

Ludwig Danzer asked the question; it is listed as a problem of Danzer in the proceedings of a convexity colloquium held in Copenhagen in 1965. The question is open. Bambah and Woods showed in 1971 that a finite union of lattices is never a Danzer set. They also constructed Danzer sets with at most Crdlog⁡d−1rC r^d \log^{d-1} r points in a ball of radius rr, by overlaying rectangular grids of different shapes. Solomon and Weiss gave a construction with O(rdlog⁡r)O(r^d \log r) points, and they showed that the vertex sets of substitution tilings, such as Penrose tilings, are not Danzer sets.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptConvex body, concept, no evidence levelConvex bodyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectDanzer set, problem, no evidence levelDanzer setproblem
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. Convex body concept
  3. Euclidean space object
  4. Outer measure concept
  5. Lebesgue measure object
  6. Danzer set problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Danzer_set
About this page
TagMMHWS2TC2A
URIm2://MMHWS2TC2A
Kindproblem
WritingW0 stub
MSC52C17, 52A37, 52C23
StatusThis copy of the site has no build data. Run m2 status MMHWS2TC2A.
AboutConvex body
UsesConvex body, Lebesgue measure
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