The Kakeya conjecture states that Besicovitch sets, sets of points in Euclidean space containing a unit line segment in every direction, must have Hausdorff dimension equal to the dimension of the space they sit in; it remains open for dimensions greater than 3. The underlying question of how small such a set can be was first asked by Soichi Kakeya in 1917. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA subset of n-dimensional Euclidean space that contains a unit line segment in every direction must have Hausdorff dimension n. 1 Proposed Year1917 is when Kakeya first asked the underlying question of how small such a set can be; the Hausdorff-dimension conjecture itself is a later formalization with no single clean statement year given in the source. Progress Toward ResolutionProved for n = 1 (trivially) and n = 2 (Davies). A proof for n = 3 by Hong Wang and Joshua Zahl was posted on arXiv in February 2025. The conjecture remains open for dimensions greater than 3. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Kakeya Conjecture (Wikipedia)
Wikimedia FoundationLead section
first asked by Soichi Kakeya in 1917
Kakeya conjecture section
that contains a unit line segment in every direction must have Hausdorff dimension n
Results section, low dimensions sentence
The Kakeya conjecture is true for n = 1 (trivially) and n = 2 (Davies).
Results section, Wang and Zahl sentence
In February 2025, a proof for the case n = 3 was posted on arXiv by Hong Wang and Joshua Zahl.
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