The Kakeya problem for codimension-two planes in ℝⁿ

@misc{lai2026kakeya,
  title = {{The Kakeya problem for codimension-two planes in $\mathbbR^n$}},
  author = {Chun-Kit Lai and Yixuan Pang and Rahul Sethi and Chenjian Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10603},
  eprint = {2610.10603},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10603},
}

TY  - UNPB
TI  - The Kakeya problem for codimension-two planes in ℝ^n
AU  - Chun-Kit Lai
AU  - Yixuan Pang
AU  - Rahul Sethi
AU  - Chenjian Wang
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10603
UR  - https://arxiv.org/abs/2610.10603
AB  - We study the (n−2)-plane Kakeya problem in ℝ^n (n≥4) subject to convex Wolff axioms. In other words, we study finite families of δ-neighborhoods of (n−2)-plane segments (δ-flakes) in ℝ^n, with the property that not too many of them can be contained inside a common convex set. Our main result is that the union of δ-flakes from such a set must have almost maximal volume. This can be viewed as a higher-dimensional generalization of Wang and Zahl's work for δ-tubes in ℝ^3. The latter implies the Kakeya set conjecture in ℝ^3. As a consequence, we resolve Rogers' $2$-plane Kakeya problem in ℝ^4 subject to certain curvature conditions on two-dimensional families of direction subspaces in the Grassmannian Gr(4,2).
ER  -