Problem dossier · Geometry & Packing
Borsuk's problem in low dimensions
Can every bounded set of diameter 1 in R^d be partitioned into d + 1 pieces of strictly smaller diameter?
§1
Status
True for d ≤ 3, false for d ≥ 64 (Kahn–Kalai-style counterexamples, refined since); every dimension from 4 to 63 is open.
Think you can crack this one? Read the playbook before you announce →
§2
The Angle of Attack
Explore candidate counterexample constructions (two-distance sets, strongly regular graph geometries) in moderate dimensions; build partition certificates for concrete bodies in R^3–R^4.
Tags: diameter · partitions · convex geometry · counterexample
§3
The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
§4
The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
§5
Related Problems
More open problems in Geometry & Packing and adjacent territory.
- Hadamard's maximal determinant problem — Geometry & Packing
- Kissing numbers in higher dimensions — Geometry & Packing
- Square packing in a square — Geometry & Packing
- Tammes problem — Geometry & Packing
- Goemans' unsplittable-flow cost conjecture — Graph Theory
- Graffiti conjecture 284 — Graph Theory