Midpoint convexity from bounded tree potentials and path costs

Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound $\hat{\delta }(t)\ge \sqrt{1+t^2/4}-1$ for $0<t<1$, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.