Critical configurations of solid bodies and the morse theory of MIN functions

Oleg V. Ogievetsky, Semen B. Shlosman

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1 Citation (Scopus)


This paper studies the manifold of clusters of non-intersecting congruent solid bodies, all touching the central ball B ⊂ R3 of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. The notion of critical cluster is introduced, and several critical clusters of balls and of cylinders are studied. In the case of cylinders, some of the critical clusters here are new. The paper also establishes criticality properties of clusters introduced earlier by Kuperberg [7].

Original languageEnglish
Pages (from-to)631-657
Number of pages27
JournalRussian Mathematical Surveys
Issue number4
Publication statusPublished - Aug 2019


  • Configurations of balls
  • Configurations of cylinders
  • Connected components
  • Critical clusters
  • Flexible clusters
  • Galois symmetries
  • Maxima of non-analytic functions
  • Platonic configurations
  • Rigid clusters


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