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

Larry Guth

Abstract

Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case.

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

Abstract

Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case.

Paper Structure

This paper contains 8 sections, 12 theorems, 34 equations.

Table of Contents

  1. The polynomial ham sandwich theorem
  2. Directed volume
  3. The proof of Theorem 1
  4. The Lusternik-Schnirelmann vanishing lemma
  5. The visibility lemma
  6. Multilinear Kakeya estimates
  7. Box estimates for unions of tubes
  8. Appendix: The Lusternik-Schnirelmann vanishing lemma

Key Result

Theorem 1

Suppose we have a finite collection of cylinders $T_{j, a} \subset \mathbb{R}^n$, where $1 \le j \le n$, and $1 \le a \le A$ for some integer $A$. Each cylinder has radius 1. Moreover, each cylinder $T_{j,a}$ runs nearly parallel to the $x_j$-axis. More precisely, we assume that the angle between th Then $Vol(I) \le C(n) A^{\frac{n}{n-1}}$.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem
  • Lemma 5.1
  • Lemma 5.2
  • Proposition
  • Theorem
  • ...and 2 more