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Pyramidal central configurations and perverse solutions. (English) Zbl 1085.70013
Summary: For $n$-body problems, a central configuration (CC) plays an important role. In this paper, we establish a relation between the spatial pyramidal central configuration (PCC) and the planar central configuration. We prove that the base of PCC is also a CC, and we also prove that for some given conditions a planar CC can be extended to a PCC. In particular, if the pyramidal central configuration has a regular polygon base, then the masses of base are equal and the distance between the top vertex and the base is fixed and the mass of the top vertex is selective. Furthermore, the pyramidal central configuration gives rise to an example of a perverse solution in ${ℝ}^{3}$.
##### MSC:
 70F10 $n$-body problems
regular polygon