Math Conjectures

Danzer's Problem

OPENLandmarkOpen problemProposed c. 1965 · Standard version

Canonical statement

Does there exist a locally finite set S⊂R2S\subset\mathbb R^2 and a constant C>0C>0 such that every convex set of area 11 meets SS, while
∣S∩[−R,R]2∣≤CR2for every R≥1? |S\cap[-R,R]^2|\leq CR^2 \qquad\text{for every }R\geq1?
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Does there exist a locally finite set
\(S\subset\mathbb R^2\) and a constant \(C>0\) such that every convex
set of area \(1\) meets \(S\), while
\[
  |S\cap[-R,R]^2|\leq CR^2
  \qquad\text{for every }R\geq1?
\]
DG-009:[Danzer1965][SolomonWeiss2016Danzer][AdiceamSolomonWeiss2020Danzer]
Discrete geometryCombinatorics

Notes

Danzer's problem, formulated around 1965 and attributed to Ludwig Danzer, asks whether there is a locally finite set S⊂R2S\subset\mathbb R^2 of bounded density — at most CR2CR^2 points in the centered square of side 2R2R, for a fixed CC — that meets every convex set of area 11 [Danzer1965]. Such a set would be dense enough to pierce every convex region of unit area, yet asymptotically no denser than a lattice.

Partial progress runs in both directions. Constructions are known whose density exceeds the lattice order only by slowly growing factors, and sets of genuinely bounded density are known to pierce restricted families of convex bodies. Solomon and Weiss studied Danzer sets alongside the related notion of dense forests [SolomonWeiss2016Danzer], and methods from homogeneous dynamics have been brought to bear on structured candidates such as cut-and-project quasicrystals [AdiceamSolomonWeiss2020Danzer].

No planar Danzer set of bounded density is known, nor is there a proof that none exists; the problem is open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.