An Equal-Edge Dispersion Bound for Planar Signed Point Configurations

We consider two disjoint classes of at least three points in the Euclidean plane carrying opposite signs, with coinciding centroids. Assuming that the signed homogeneous potential of positive exponent vanishes and that one same-sign distance equals one opposite-sign distance, we establish a strict lower bound on the ratio of the largest to the smallest remaining same-sign distances after removal of the matched pair. The threshold is defined as the minimum of a finite family of uniquely determined positive roots of explicit scalar equations. No incidence assumption is imposed on the two equal-length edges. The result applies in particular to collision-free critical points of the signed potential and therefore to balanced signed equilibria. The estimate concerns a residual within-class subfamily of distances and is complementary to earlier global bounds on the full set of mutual distances for zero-total-mass systems. Aggiungi alla chat Traduci in italianoSpiega meglio il concetto di equilibrioRiscrivilo in un inglese più semplice