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3D Kakeya problem (AlphaEvolve problem 10)

3D Kakeya problem (AlphaEvolve problem 10). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 28A75 · Length, area, volume, other geometric measure theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let n≥2n \geq 2. Let C(n)C(n) denote the minimal volume ∣⋃j=1n⋃k=1nPj,k∣|\bigcup_{j=1}^n \bigcup_{k=1}^n P_{j,k}| of prisms Pj,kP_{j,k} with vertices

(xj,k,yj,k,0),(xj,k+1n,yj,k,0),(xj,k,yj,k+1n,0),(xj,k+1n,yj,k+1n,0), (xj,k+jn,yj,k+kn,1),(xj,k+j+1n,yj,k+kn,1),(xj,k+jn,yj,k+k+1n,0),(xj,k+j+1n,yj,k+k+1n,1)\begin{align*} &\quad (x_{j,k},y_{j,k},0), \left(x_{j,k}+\frac{1}{n},y_{j,k},0\right), \left(x_{j,k},y_{j,k}+\frac{1}{n},0\right), \left(x_{j,k}+\frac{1}{n},y_{j,k}+\frac{1}{n},0\right),\ &\left(x_{j,k}+\frac{j}{n},y_{j,k}+\frac{k}{n},1\right), \left(x_{j,k}+\frac{j+1}{n},y_{j,k}+\frac{k}{n},1\right), \left(x_{j,k}+\frac{j}{n},y_{j,k}+\frac{k+1}{n},0\right), \left(x_{j,k}+\frac{j+1}{n},y_{j,k}+\frac{k+1}{n},1\right) \end{align*}

for some real numbers xj,k,yj,kx_{j,k}, y_{j,k}. Establish upper and lower bounds for C(n)C(n) that are as strong as possible.

Lean 4

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Background

Repository result category: world record — AlphaEvolve improved on the best known result recorded at the time. See Section 6.5 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — 3D Kakeya problem (AlphaEvolve problem 10) linkApache-2.0 (repository license); problem pages reproduced with attribution.

Recorded progress

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