Monotonicity of edge-use regions in the cost parameters

Determine whether, for most edges in the minimal matchings generated by the unbalanced cost functions, the set of parameter pairs $(\gamma,a)$ for which a given edge is used is monotone in $a$ and in $\gamma$.

Background

The paper studies minimal matchings of two independent Poisson point processes on the real line under the unbalanced costs $f_{\gamma,a}$, where $\gamma$ controls the distance exponent and $a$ biases the cost of blue-red edges relative to red-blue edges. Numerical diagrams suggest that, as these parameters vary, individual edges generally enter or leave the matching in an ordered fashion rather than reappearing after disappearing.

The authors formulate this observed behavior as a belief concerning the parameter region associated with each edge. Establishing monotonicity would provide a more systematic description of how unbalanced minimal matchings change across parameter space and would strengthen the qualitative conclusions drawn from the numerical experiments.