On the realization space of the cube

Karim Adiprasito, Daniel Kalmanovich, Eran Nevo

Research output: Contribution to journalArticlepeer-review


We prove that the realization space of the d-dimensional cube is contractible. For this we first show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. As an application we use this fact to define an analog of the connected sum construction for cubical d-polytopes, and apply this construction to certain cubical d-polytopes to conclude that the rays spanned by f -vectors of cubical d-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further it shows that the respective realization spaces are contractible.

Original languageAmerican English
Pages (from-to)261-273
Number of pages13
JournalJournal of the European Mathematical Society
Issue number1
StatePublished - 2024

Bibliographical note

Publisher Copyright:
© 2023 European Mathematical Society.


  • Cubical polytopes
  • connected sum
  • face numbers
  • realization space


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