Plane with A∞-weighted metric not bilipschitz embeddable to Rn

A planar set G subset of R-2 is constructed that is bilipschitz equivalent to (G, d(z)), where (G, d) is not bilipschitz embeddable to any uniformly convex Banach space. Here, z epsilon (0, 1) and d(z) denotes the zth power of the metric d. This proves the existence of a strong A(infinity) weight in R-2, such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of R-2.