Hi,

As far as I understand your question, a partial answer may be recovered from the Khachiyan's original covergence proof of the ellipsoid algorithm. Namely, Khachiyan used the volume as an intermadiate parameter and expressed it via, what he called, the thickness $r(E)$ that is equal to $\lambda_{min}$ of the current ellipsoid $E$. The following inequality holds $r(E_{next})\geq \frac{d}{d+1} r(E_{previous})$. And $\lambda_{max}$ can be upperbounded by the inequality $\lambda_{max-next}\leq 2^{\frac1{d^2}}\lambda_{max-previous}$. Thus, the condition number increases not faster than $[\frac{d}{d+1} 2^{\frac1{d^2}}]^n$. It seems that you can check this exponential rate in your toy example (just consider the plane case $d=2$ to simplify the computations). By the way, precisely the fact that the ellipsoid algorithm USUALLY operates according to the theoretical estimates is a main reason for claims of it's ``practical'' impracticallity. But it's another story.

Sergey