For a cup of coffee more

I was listening to a marvelous talk by Jean-Pierre Bourguignon today, about generalizations and (geo)metric meaning of scalar curvature, and Misha Gromovs thoughts on the subject. Essentially, just like Alexandrov and Gromov and many others gave us metric understanding of sectional curvature, and Sturm and Lott-Villani and many more gave us metric ways of thinking about Ricci curvature, Misha is at it again to do the same with the weakest notion of curvature.

One of the goals would be to understand polyhedral spaces from the viewpoint of scalar curvature. In this context, Misha asked me to give a reference for the following statement: Given fixed and any , there is a finite number of combinatorially distinct -polytopes all whose dihedral angles are all smaller than I will write more about this and scalar curvature later, but for now I am stealing a format from Gil’s blog and ask: Did I have a chance to find such a reference, or do you find examples that make such a statement hopeless? And what can you choose safely?

While I am at it, let me add another question for your intuition: Can you deform a polytope such that all dihedral angles do not get bigger? What deformations can you find? More on this subject, and some partial answers to the questions above, later.