Dvoretzky's theorem states that each centered convex body of sufficiently high dimension has an "almost
spherical" 
-dimensional
central section.
Klartag and Moshe (2026) proved a polynomial bound in the approximation parameter. Let 
be a convex body containing the origin
in its interior, with 
and 
. If
 |
for a universal constant 
, then there are an 
-dimensional linear subspace 
and 
such that
 |
where 
is the unit ball in 
. This resolves the conjecture that the required ambient dimension has polynomial dependence on 
for fixed 
.
Klartag and Moshe (2026) state that the choice of probability distribution used in the proof arose from a discussion between ChatGPT and Moshe. ChatGPT was also used to inspect and improve proofs.
