Maximum box volume in an ellipsoid ($V_{\max }=8abc/(3\sqrt{3})$)

For an axis-aligned box, maximize $xyz$ subject to $x^2/a^2+y^2/b^2+z^2/c^2=1$. The Lagrange multiplier equations imply $x^2/a^2=y^2/b^2=z^2/c^2=1/3$. Boundary boxes have zero volume, so this gives the maximum. Arbitrary orientation cannot improve it: averaging the ellipsoid inequality over the eight vertices gives a sum of three squared transformed half-edge lengths at most one; the determinant bound and arithmetic-geometric mean bound their product by $1/(3\sqrt{3})$.