Generalized closing theorems (Q654726)

Closing theorems, in particular the theorem of Poncelet, are results of projective geometry. The author formulates and proves a generalized closing theorem for families of Euclidean spheres in \(\mathbb{R}^d\). A corollary of this theorem gives a general closing principle for spheres in \(\mathbb{R}^3\), from which the classical theorems in \(\mathbb{R}^2\) of Poncelet, Steiner and Emch, and the zigzag theorem can be derived. Further results are the extension of the above mentioned classical theorems to \(\mathbb{R}^d\) and an alternative, elementary proof of the Emch therorem.