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in the online article A formula for the N-circumsphere of an N-simplex dated April 2013, G. Westendorp provides an interpretation of the entries of inverse of Cayley-Menger matrices $\hat{B}$, that resulted in an efficient solution to my problem Determining the Largest Face of a Simplex, contrary to what anyone (including me) would have expected.

The reason for this question is that despite the really amazing properties of $\hat{B}$$^{-1}$, I couldn't find any other ressource discussing the properties of $\hat{B}$$^{-1}$. Even more astonishing is the fact, that there is a later article A Recursive Formula for the Circumradius of the n-Simplex from 2016, that discusses the calculation of the circum-radius of a simplex via a recursive formula, which isn't even proved correct for arbitrary dimensions.

Provided the correctness of the results that are presented in Westendorp's article, the result of the later publication would be inferior and I conclude that Westendorp's interpretation of $\hat{B}$$^{-1}$ is either largely unknown or not correct (what I don't believe).

Question:

  • have there already been results published about the entries of $\hat{B}$$ ^{-1}$ prior to Westendorp's article(s)?
  • have Westendorp's results been noticed by mathematicians concerned with the properties of simplices?
  • where can I find other ressources about the properties of $\hat{B}$$^{-1}$?


Remark: the notation $\hat{B}$ for the Cayley-Menger matrix is adapted from WolframAlpha and addresses the feedback of @YCor

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  • $\begingroup$ @YCor the edit is fine with me; I took my "notation" from Westendorp's article. Lets see if others also have suggestions for better notation. What do you think about adding periods like C.M.$^{-1}$? $\endgroup$
    – Manfred Weis
    Commented Mar 3, 2018 at 12:56
  • $\begingroup$ I'm happy with the new acronym (I think it's unnecessary in the title, by the way) $\endgroup$
    – YCor
    Commented Mar 3, 2018 at 14:57
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    $\begingroup$ Note, there is an interesting direct recursive approach (I think maybe it harkens back to Laserre in '83?) math.stackexchange.com/questions/4056099/… ... have been trying to understand this as some (possibly optimal in some sense) inversion algorithm for the CM matrix. @ManfredWeis would be interested in anyone has come across analysis of algorithmic solutions (the compute graph) of the inversion problem. The Recursive solution is quite appealing as a proof structure as well I think. $\endgroup$
    – safetyduck
    Commented Mar 10, 2022 at 11:45

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