Borsuk's conjecture fails in dimension nine

The explanation

Draw an equilateral triangle with sides of length 1. Its diameter, the largest distance between two of its points, is 1. Now cut it into pieces, each with a diameter smaller than 1. Two pieces cannot work. One piece would get two corners, and those sit 1 apart. Three pieces do work. Cut from the center to the middle of each side. Each piece is then only about 0.58 across.

Karol Borsuk asked in 1933 whether this is the worst that can happen. Take any bounded set in d-dimensional space. Can it always be cut into d + 1 pieces of smaller diameter? The triangle shows that the plane, with d = 2, sometimes needs all three. This paper builds a set in nine-dimensional space that ten pieces cannot handleSource 1. It is a counterexample, a single case where the general claim fails.

The set is built from lines. Take a line through the center of four-dimensional space. Pick a direction u along it, of length 1. Form a 4-by-4 table whose entry in row i and column j is u_i times u_j. Flipping u to the opposite direction gives the same table. So each line becomes exactly one table, called its projectorSource 2. The distance between two tables comes from their 16 entries. Square each difference, add them up, and take the square root.

Here are real numbers. The lines along the first two axes give tables that differ in two entries, each by 1. Their distance is the square root of 2, about 1.414. The line along (1, 1, 0, 0) gives a table with 1/2 in a 2-by-2 block at the top left. Its distance from the first axis is exactly 1. The line along (1, 1, 1, 1) sits about 1.22 from the first axis. The paper gives the general rule. The squared distance is 2 minus twice the squared cosine of the angle between the linesSource 3. So two tables are as far apart as possible, about 1.414, exactly when their lines meet at a right angleSource 4.

Why nine dimensions? A symmetric 4-by-4 table is fixed by its 10 entries on or above the diagonal. For every projector, the diagonal entries add up to 1. That one condition leaves a flat space of nine dimensions, with ordinary distancesSource 5. Borsuk's question therefore allows ten pieces. The paper proves that ten pieces of diameter below about 1.414 can never cover all the projectorsSource 1.

The record before this was far higher. Jeff Kahn and Gil Kalai first disproved Borsuk's claim in 1993, in very high dimensions. Later work brought the dimension down to 946, then 561, then 560. Methods based on spherical codes, carefully placed points on a sphere, reached 323, then 321, then 298 by 2003. Andriy Bondarenko reached 65 in 2014, and Thomas Jenrich and Andries Brouwer reached 64. A 2026 note by Max Grinsztajn gives a finite set of points in dimension 63. The projector set itself was already known to experts. What is new is the proof that ten pieces failSource 6.

The hard part is that the set is infinite and smoothly curved. Its shape is the space of all lines through the center of four-dimensional space. Suppose ten good pieces existed. The paper turns them into ten smooth weights on the lines, adding up to 1Source 7. Two lines at right angles never share a positive weight. The paper records which groups of weights are positive together. Known facts about this space concern tidy triangulations, meshes of triangles and their higher versions. The recorded pattern is not one, so those facts do not apply directlySource 8. The paper extracts enough structure from the weights themselves, using topologySource 9. A finite argument about the ten labels then gives a contradictionSource 10.

So split all these lines into any ten groups you like. Some group will always stretch the full 1.414 across. For the triangle, one more piece fixed the problem. Here the paper does not say whether eleven pieces suffice. It does not find the exact number of pieces this set needsSource 11. It also does not find the smallest dimension where Borsuk's claim fails. It does claim to extend the example to every dimension from 9 upward, by adding extra pointsSource 12.

How sure is this? A computer proof checker called Lean has checked a formal version of the main theorem. That statement covers the nine-dimensional projector set, its diameter, and the failure of ten arbitrary pieces. The Lean scope note does not mention the extension to higher dimensions. No specialist review of the paper has been published.

What it doesn't show. This does not find the smallest dimension where Borsuk's conjecture fails, or the exact number of pieces the set needs.