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I have a problem, that requires sorting a set of simplices, that are defined via their sidelengths, according to volume; the value of the individual volumes isn't relevant in my problem.

Question:

are there faster methods of comparing the volumes of two simplices (that are defined via their sidelengths), than comparing the absolute values of their Cayley Menger determinants?

  • if no, where can I find details about the initial proof?
  • if yes, what are relevant algorithms for that problem?
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    $\begingroup$ Seems unlikely there is anything better than the determinant calculation. But at least there has been some work on optimizing the calculation: Kahan: Volume of a Tetrahedron PDF download. $\endgroup$
    – Joseph O'Rourke
    Commented Feb 17, 2018 at 11:37
  • $\begingroup$ @JosephO'Rourke I also don't put much hope in the existence of something better, and I guess I will have to accept an $O(n^5)$ time complexity for the problem I am trying to solve (canonical ordering of points on basis of distance measurements). Thanks for the pointer to Kahan's paper anyways; that reduces the risk of wrong ordering because of numeric issues.. $\endgroup$
    – Manfred Weis
    Commented Feb 17, 2018 at 11:55

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