We know that, given an n-dimensional Euclidean simplex, for all $1\leq i,j,k,l\leq n+1$, we have(law of sines)$$\frac{A_i A_j}{A_k A_l}=\frac{c_{ij}}{c_{kl}}$$(from Elementary Formulas for a Hyperbolic Tetrahedron) And in http://www.emis.de/journals/JIPAM/images/106_03_JIPAM/106_03.pdf it mentioned some inequalities for simplexes in Euclidean space.

I wonder if there is a formula about law of sines and cosines for hyperbolic n-simplexes which only have terms: side lengths, area, dihedral angles.



Yes, something like that is proved in the paper by Simon Kokkendorff:

Kokkendorff, Simon L., Polar duality and the generalized law of sines, J. Geom. 86, No. 1-2, 140-149 (2006). ZBL1115.51010.

It would be too cumbersome to define everything here, but the paper is very nicely written.


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