Synthetic condition for metric spaces related to MTW condition.

We introduce a new condition for (k+l+2)-point subsets in metric spaces and call it (k,l)-dipole property. For intrinsic metric space this condition is more strong then nonnegative curvature condition and holds naturally for quotients of Euclidean spaces  by isometric group action. It holds also for biquotient spaces of Lie groups with bi-invariant metric.

We show that (3,1) and (2,2)-dipole properties hold for any CBB(0) space while (3,3)-dipole property doesn't.  The first two give new conditions for 6-point subsets in CBB(0) space and the last gives a condition which allows to see the difference between subsets of CBB(0) space and subsets of quotients of Euclidean spaces.

We also prove that (4,1)-dipole property implies Maהrudingerחang (MTW) condition. MTW condition is a 4th-order
nonlocal curvature condition (more strong then nonnegative sectional curvature) related to the regularity of optimal transport. This gives a hope to prove  continuity of optimal transport without using a smooth structure. As an application we note that biquotient spaces of Lie groups with bi-invariant metric satisfy MTW condition.

This is joint work with A. Petrunin and V. Zolotov.