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?

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?