The distance set of a subset 
of Euclidean space is the
set of distances between pairs
of its points,
where 
is the Euclidean norm. The Falconer
distance conjecture relates the Lebesgue measure
of this set to the Hausdorff
dimension of 
.
Lower bounds on 
alone do not establish positive Lebesgue measure
(Iosevich and Yavicoli 2026).
See also
Distance,
Falconer
Distance Conjecture,
Hausdorff Dimension
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References
Iosevich, A. and Yavicoli, A. "Improved Dimension Bounds for Falconer's Distance Problem in the Plane." 4 Oct 2026. https://arxiv.org/abs/2610.05500.OpenAI.
"The Falconer Distance Conjecture in All Dimensions." Sep. 23, 2026a.
https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf.
Cite this as:
Weisstein, Eric W. "Distance Set." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DistanceSet.html
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