[ i sent a similar message several days ago, but it appears to
have gotten lost. my apologies if anyone already got this. ]
in the same way, local maxima (within the antislice group) in the
90 degree antislice metric are local maxima in the full cube group
(quarter turn metric).
this isn't necessarily true. one must check that the minimal
maneuvers (within the antislice group) for such positions are
also minimal in the full group.
the position i mentioned
(FB') (RL') (U'D) (R2L2) = [ ... ]
is quickly checked to require 10 quarter turns, so indeed it
is locally maximal.
here's an example i found of a locally maximal position whose
inverse is not locally maximal:
A = B U2 F2 R U' R' B' R' U F2 U2 (15q)
this position has six symmetries, generated by the cube rotation
C_UFR and central reflection. using these symmetries we can
give minimal maneuvers which end with a half turn of any face,
and thus with any of the twelve quarter turns. the same is not
true of its inverse, and we can easily check that there is no
minimal maneuver for A which begins with the quarter turn B'.
equivalently, the position A^-1 B' requires 16 quarter turns.