Abraham Wald and the volume of the 4-simplex

Abraham Wald is one of the figures who keeps popping up in HOW NOT TO BE WRONG.  I just learned yet another cool thing about him which doesn’t have a place in the book, so I’m putting it here.

You know Heron’s formula, which gives you the area of a triangle in terms of the lengths of its edges.  And there’s a generalization:  the Cayley-Menger determinant is a formula for the volume of an n-simplex in terms of its edge lengths.

What about other kinds of faces?  The volume of a tetrahedron is determined by the six edge lengths, but not by the areas of its four faces.  Imagine four identical isosceles right triangles formed into a flattened tetrahedron that’s really just one square laid on top of another; that tetrahedron has volume 0, but a regular tetrahedron with faces of the same area obviously has positive volume.

What about a 4-simplex?  The lengths of the 10 edges determine the volume.  So you might guess that the areas of the 10 2-faces would give you a way to compute the volume, too.  But nope!  Wald gave an example of two 4-simplices with the same face-areas but different volume.  I wonder what the space of 4-simplices with fixed face-areas looks like.

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One thought on “Abraham Wald and the volume of the 4-simplex

  1. NDE says:

    Generically the space must be finite, right? The edge lengths identify the space of 4-simplices with an open set in R^10, so the map from this space to the 2-face areas must be generically finite-to-one unless the 2-face areas satisfy some polynomial identity (surely that can’t be true but I’ve not checked it – should be enough to pick a random 4-simplex and perturb it infinitesimally). There might still be interesting positive-dimensional fibers, though.

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