  We argue that in clean high-$\kappa$ type II superconductors, the low
temperature vortex core size (defined as the coherence length $\xi$) in high
fields should decrease with increasing applied field in qualitative agreement
with experimental data. Calculations are done for the Fermi sphere and cylinder
(with the field parallel to the cylinder axis). The results for clean materials
at T=0 can be represented as $\xi(H)/\xi(H_{c2}) = U (H/H_{c2})$ with $U$ being
an universal function.
