Acta Mathematica

The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture

Larry Guth

Full-text: Open access


We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery and Tao. The proof uses the polynomial method introduced by Dvir.

Article information

Acta Math., Volume 205, Number 2 (2010), 263-286.

Received: 15 December 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2010 © Institut Mittag-Leffler


Guth, Larry. The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture. Acta Math. 205 (2010), no. 2, 263--286. doi:10.1007/s11511-010-0055-6.

Export citation


  • B ennett, J., C arbery, A. & T ao, T., On the multilinear restriction and Kakeya conjectures. Acta Math., 196 (2006), 261–302.
  • D vir, Z., On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22 (2009), 1093–1097.
  • G romov, M., Isoperimetry of waists and concentration of maps. Geom. Funct. Anal., 13 (2003), 178–215.
  • — Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles. Geom. Funct. Anal., 19 (2009), 743–841.
  • G uth, L., Minimax problems related to cup powers and Steenrod squares. Geom. Funct. Anal., 18 (2009), 1917–1987.
  • — Directional isoperimetric inequalities and rational homotopy invariants. Preprint, 2008. arxiv:0802.3549v1 [math.DG].
  • — Isoperimetric inequalities and rational homotopy invariants. Preprint, 2008. arxiv:0802.3550v1 [math.DG].
  • H atcher, A., Algebraic Topology. Cambridge University Press, Cambridge, 2002.
  • H urewicz, W. & W allman, H., Dimension Theory. Princeton Mathematical Series, 4. Princeton University Press, Princeton, NJ, 1941.
  • K atz, N. H., Ł aba, I. & T ao, T., An improved bound on the Minkowski dimension of Besicovitch sets in R3. Ann. of Math., 152 (2000), 383–446.
  • L oomis, L. H. & W hitney, H., An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc, 55 (1949), 961–962.
  • M atoušek, J., Using the Borsuk–Ulam Theorem. Universitext. Springer, Berlin–Heidelberg, 2003.
  • S tone, A. H. & T ukey, J. W., Generalized “sandwich” theorems. Duke Math. J., 9 (1942), 356–359.