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This question is directly inspired by Pietro Majer's question and my answer to it.

One can define a simplex, and the dihedral angles thereof in an infinite dimensional Hilbert space (one has to take some care that all of the faces are actually closed subspaces -- for example, the affine subspace spanned by an orthonormal basis is not a good codimension-one subspace). The question is then: what dihedral angles are possible? Suppose we now postulate that all the dihedral angles are equal. How many simplices are there satisfying that condition (the possible answers can be none, one, seventeen, infinitely many...)? Since the answer I give (which is similar to what @Pietro had in mind) is inductive, it does not work at all in this setting.

Of course, I don't know from infinite dimensional geometry, so this question could be totally stupid.

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