Link between the decomposability bundle and Wasserstein tangent cones for alternative transport costs

Determine whether, for a transport cost function different from the quadratic cost |x−y|^2, the Alberti–Marchese decomposability bundle D^AM of a measure μ can be linked to (or coincide with) the Grassmannian section that characterizes the centred tangent cone in the corresponding Wasserstein geometry. Specifically, identify a transport cost under which D^AM aligns with the Grassmannian section arising from the Wasserstein tangent cone.

Background

The decomposability bundle D^AM, introduced by Alberti and Marchese, encodes directions along which a measure decomposes into rectifiable components. In the present work, the Grassmannian section D^Sol that characterizes centred solenoidal (and dually tangent) cones in the 2-Wasserstein geometry can diverge from D^AM, as illustrated by examples where D^Sol equals the full space while D^AM yields classical tangents.

This discrepancy motivates asking whether a different transport cost could reconcile the two structures, making the Wasserstein tangent cone’s Grassmannian section coincide with D^AM. Clarifying this relationship would bridge geometric measure theory with optimal transport under generalized costs.