# Maximum volume inscribed ellipsoid of given aspect ratio

It is well know (see Boyd & Vandenberghe's Convex Optimization, Sec. 8.4.2) that the maximum volume ellipsoid inscribed in a given convex polytope in $$\cal{H}$$-form can be computed by solving the Semi-Definite-Program

$$\mbox{max} \qquad \log \det B$$

$$\mbox{s.t.} \qquad \vert\vert B a_i\vert\vert_2 + a_i^{\rm T} d \leq b_i, \quad B \succcurlyeq 0$$

I am looking for a way to add to this problem contraints on the ratios of the ellipsoid semi-axes. In other words, if $$\lambda_i, i=1,\ldots,n$$ are the eigenvalues of $$B B^{\rm T}$$,

$$r_i = \frac{\lambda_i}{\lambda_1}, \quad i=2,\ldots,n$$

It seems that if we add this new constrains the original problem is not even convex anymore. Any ideas?