Midpoint convexity from two recursive potentials
We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least $t^3/128$ for $0\lt t\lt 1$, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.
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