Adaptive local mesh-size conjecture for penultimate histograms

Let $PEN$ be a finite set of interior histograms and, for each $y\in PEN$, define the local mesh size $LMS(y)$ by

$$LMS(y)=\min_{x\in PEN-\{y\}}\lVert x-y\rVert.$$

For a target histogram $G\in\mathcal{H}^{\circ}$, let $h(y,G)=\lVert\nabla_y C(y,G)\rVert$ be the norm of the cost gradient. Adaptive mesh-size conjecture. Sparse but efficient finite sets $PEN$ of penultimate points should satisfy

$$LMS(y)\simeq \frac{c}{h(y,G)},$$

where $c$ is a constant. This is a numerical proposal for reducing the number of penultimate points in multi-stage geodesic searches; its validity is presented as a pragmatic conjecture and is not established by the paper.