Property (G) in UCW-hyperbolic spaces

Determine whether every UCW-hyperbolic space—namely, every W-hyperbolic space that admits a monotone modulus of uniform convexity—has property (G) with some modulus ψ. Concretely, establish whether for all r, ε > 0 and all points a, x, y with d(x,a) ≤ r, d(y,a) ≤ r, and d(x,y) ≥ ε, the inequality d((x+y)/2,a)^2 ≤ (d(x,a)^2 + d(y,a)^2)/2 − ψ(r,ε) holds in all UCW-hyperbolic spaces.

Background

The paper introduces property (G) for W-hyperbolic spaces: a quantitative midpoint inequality with a modulus ψ that strengthens uniform convexity-type behavior and is satisfied by CAT(0) spaces (Proposition 2.7).

UCW-hyperbolic spaces are W-hyperbolic spaces equipped with a monotone modulus of uniform convexity. While CAT(0) spaces are known to satisfy property (G), the authors highlight that it is not known whether this stronger property holds for UCW-hyperbolic spaces in general.

To ensure the applicability of Reich’s theorem in broader nonlinear settings, the authors identify a weaker condition, property (M), which suffices for their main results; however, establishing property (G) for UCW-hyperbolic spaces would yield a cleaner and stronger framework.