Do you demand that the polyhedron be convex? Otherwise, there's a really boring $k = 1$ solution (for the weaker notion of convergence): rasterize the sphere by approximating it as the union of many tiny cubes, and then erect a shallow square-based pyramid on every square face that results.

Even Steven J. Miller's CV is in LaTeX: web.williams.edu/Mathematics/sjmiller/public_html/math/… which suggests that even if he did sign off that affidavit, he didn't write the statistical analysis himself.

or the three elementary symmetric polynomials in the lengths of the sides?

A good counterexample is $\beta \mathbb{N}$ and its subset $\beta \mathbb{N} \setminus \mathbb{N}$.

I'm afraid I can't answer the comment [about the 46-variable quadratic form]: the runtime will depend on both the quadratic form itself (not just the number of variables) and the solver you're using, so the best way to answer that question is to experiment yourself.

@mathworker21 Equation (5) contains a nonlinear constraint $x^T \lambda = 0$ (because neither $x$ nor $\lambda$ is a constant), so it's not a MILP. Equation (6) has purely linear constraints and objective, so it's in the correct form for you to input it into a MILP solver.

Thanks! Yes, it looks like the space is called DIII in Cartan's classification of compact Riemannian symmetric spaces. (I'll accept this if you post it as an answer.)