Gale's conjecture asserts that among bounded sets in 
-dimensional
Euclidean space, for every 
, the ball requires the largest
relative generalized diameter when partitioning
into 
parts (Voronov 2026). In three dimensions, it predicts
that every bounded set can be partitioned
into four parts, each of generalized diameter
at most 
times the generalized diameter of the original
set. The factor is 
and is optimal for a sphere
(Voronov 2026). This is a quantitative strengthening of the three-dimensional case
of Borsuk's conjecture.
Voronov (2026) constructed a set 
of 101 points in 
such that every set partition
of 
into four parts has a part of generalized diameter
greater than 
.
This refutes Gale's conjecture without refuting Borsuk's
conjecture in three dimensions. Voronov (2026)
reports that GPT-6 Astra helped develop the search algorithms
and heuristics, including ideas proposed by the model.
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20 Oct 2026.