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
Much of the research in this area has studied the problem of how small such sets can be, first asked by SÅichi 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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