0

In the book by Bridson and Häfliger, the following question is presented:
Prove that the identity map of any abstract simplicial complex induces a bi-Lipschitz homeomorphism between any two regular $M_\kappa$-simplicial complexes associated with $K$ (where $\kappa$ is not fixed).
I am interested in the case where the map goes between a possibly infinite-dimensional euclidean simplex and its spherical variant. The induced map in question would then be the projection from the simplex to the unit sphere. It is easy to see that the Euclidean distance is smaller than the corresponding spherical distance. However, I struggle to prove the other inequality, namely, that the projection is Lipschitz continuous.

segi
  • 1
  • 1
  • I'm not so familiar with simplices, but I know that, in any inner product space (or, I believe with more work, any uniformly rotund space), the projection onto the unit sphere is Lipschitz on a set if $0$ belongs to the exterior of the set. Would that be helpful? – Theo Bendit Mar 30 '24 at 18:35
  • I need the Lipschitz continuity with respect to the spherical metric. You're probably referring to Lipschitz continuity with respect to the metric coming from the inner product restricted to the sphere, correct? – segi Mar 30 '24 at 19:45
  • Indeed I am. And the identity map is not Lipschitz between these two metrics? – Theo Bendit Mar 31 '24 at 03:44
  • Actually, it is. Now, on the sphere, we can write the distances as real functions of the inner product, then because $\frac{\arccos (t)}{\sqrt{2(1-t)}}$ is decreasing, the identity map is Lipschitz. So, I suppose with the fact you mentioned, this would resolve the question. Do you have a source for that? I'm also interested in the more general case you mentioned. – segi Mar 31 '24 at 12:30
  • I've just realized you said 0 should belong to the exterior. In the infinite dimensional case, 0 lies in the closure of the simplex. Is this somehow resolvable? – segi Mar 31 '24 at 13:12
  • It's likely not resolvable. If your set contains two open line segments starting at $0$ going in two distinct directions, the projection map will not be Lipschitz. – Theo Bendit Mar 31 '24 at 17:36

0 Answers0