A Minkowski plane is a two-dimensional real normed space. Its distance is 
, and its closed unit
ball is a centrally symmetric convex set with the
origin as its center and in its interior. Conversely,
every such compact set 
defines a norm by
 |
This meaning is distinct from the indefinite metric of Minkowski space.
The chromatic number of a Minkowski plane is the least number of colors needed so that points at distance one have different colors. A circular unit
ball gives the Hadwiger-Nelson problem.
Exoo et al. (2021) proved lower bounds of five for regular octagonal, decagonal,
and dodecagonal unit balls and asked whether five is
a lower bound for every regular 
-gon with 
. Gehér (2023) gave an upper bound of six for
the regular polygons with at most 22 sides.
MathIsEvenEasier (2026) reported the lower bound five for the regular fourteen-gon. The construction has 1,540 labeled vertices and 13,755
edges of norm-distance one, and admits no four-coloring.
Combined with Gehér's bound (Gehér 2023), it gives 
. GPT-6 Astra developed the geometric Lean argument
and connected it to a kernel-checked coloring refutation. As of Oct. 8, 2026,
the public Lean build covered both the geometry and the finite obstruction, but independent
mathematical review and a definitive priority check had not been reported. The exact
value remains unknown.
