Editing Kakeya set (section)

===Statement===
The same question of how small these Besicovitch sets could be was then posed in higher dimensions, giving rise to a number of conjectures known collectively as the ''Kakeya conjectures'', and have helped initiate the field of mathematics known as [[geometric measure theory]].  In particular, if there exist Besicovitch sets of measure zero, could they also have s-dimensional [[Hausdorff measure]] zero for some dimensions less than the dimension of the space in which they lie? This question gives rise to the following conjecture:

:'''Kakeya set conjecture''': a set in Euclidean space that contains a unit line segment in every direction must have a Hausdorff dimension equal to the dimension of the space. 

This is known to be true for ''n'' = 1, 2 but only partial results are known in higher dimensions.

In February 2025, a claimed proof for the case ''n'' = 3 was posted on [[arXiv]] by [[Hong Wang (mathematician)|Hong Wang]] and Joshua Zahl.<ref name=":1">{{cite arXiv |eprint=2502.17655 |class=math.CA |author1=Hong Wang |author2=Joshua Zahl |title=Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions |date=2025-02-24}}</ref> The Kakeya conjecture in three dimensions is described as "one of the most sought-after open problems in geometric measure theory", and the claimed proof is considered to be a breakthrough.<ref>{{cite web |title=Chinese maths star Wang Hong solves 'infamous' geometry problem |url=https://www.scmp.com/news/china/science/article/3300958/chinese-maths-star-wang-hong-solves-infamous-geometry-problem?module=perpetual_scroll_0&pgtype=article |archive-url=https://archive.today/20250304114650/https://www.scmp.com/news/china/science/article/3300958/chinese-maths-star-wang-hong-solves-infamous-geometry-problem |archive-date=4 March 2025 |date=4 March 2025|publisher=[[South China Morning Post]]}}</ref><ref>{{cite web |title=Century-Old Math Enigma Finally Solved: How Chinese Student Cracked An 'Impossible' Geometry Mystery |url=https://english.jagran.com/world/chinese-student-wang-hong-cracked-an-impossible-geometry-puzzle-that-stumped-mathematicians-for-over-a-century-10221886 |website=[[Dainik Jagran]] |archive-url=https://archive.today/20250304154738/https://english.jagran.com/world/chinese-student-wang-hong-cracked-an-impossible-geometry-puzzle-that-stumped-mathematicians-for-over-a-century-10221886 |archive-date=4 March 2025 |date=4 March 2025}}</ref><ref>{{Cite web |last=Howlett |first=Joseph |date=2025-03-14 |title='Once in a Century' Proof Settles Math's Kakeya Conjecture |url=https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/ |access-date=2025-03-21 |website=Quanta Magazine |language=en}}</ref>