Under Review Elsevier · Computational Geometry
Geometry and Volume of Stepping-Stone Diversion Neighborhoods in Euclidean Spaces of Arbitrary Dimension
Summary
This manuscript develops a geometric, graph-theoretic, and probabilistic analysis of the stepping-stone graph in \(\mathbb{R}^d\) for every \(d\geq 2\). It derives a normalized one-dimensional volume integral, recovers the degenerate and Gabriel cases and the relative-neighborhood limit, proves strict region and volume monotonicity, establishes graph inclusions and connectivity, and validates a stable quadrature formula by rejection sampling.
$$ a_{d,\mathrm{SS}}(\alpha)=2\kappa_{d-1}\int_{2^{-1/\alpha}}^{1}y_\alpha(u)^{d-1}x_\alpha^{\prime}(u)\,du $$
Context
The manuscript is framed as a dimension-lift of a planar empty-region computation: symmetry reduces the d-dimensional stepping-stone region to a one-dimensional quadrature over transverse ball slices.
Main contributions
- Defines the normalized stepping-stone region \(K_{\mathrm{SS},\alpha}\) and factors its volume into pair distance and a normalized constant.
- Derives an explicit boundary parametrization and a one-dimensional integral formula for every \(d\geq 2\) and \(\alpha\geq 1\).
- Recovers the degenerate and Gabriel cases and proves convergence to the open relative-neighborhood lune.
- Establishes strict region and volume monotonicity, graph inclusions, and connectivity in arbitrary dimension.
- Gives a stable quadrature formula and validates it independently by rejection sampling.