On 18/01/2021 10:30, Bill Gosper wrote:

If a plane P through the center of an ellipsoid E intersects it in area A, the volume of E is 2Ah/3, where h is the distance between the two planes parallel to P and tangent to E.

Consider an affine map f from E to an equal-volume sphere S. Chop E up into thin slices by planes parallel to P. f is volume-preserving, so in particular preserves volumes of these slices. Each slice's area is scaled by some factor k (the same for all because the planes are parallel) and its thickness is scaled by some factor, likewise the same for all, which must be 1/k because the volumes are preserved. Now A is 1/k times the area of f(A) and h is k times the corresponding height (i.e., the diameter of the sphere), so your conjecture is true for E iff it's true for S. Which, unless you dropped a factor of 2 or something (I haven't checked), it is. -- g