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Here is Ilya's $0.821$-ellipse (if I interpreted his intention correctly), discretized to an $180$-point polygon at $2^\circ$-degree angular increments:
         
His point, if I may editorialize, is that the naive lower bound of $4/\sqrt{\pi} \approx 2.26$ is not so far from the bound for the optimal ellipse. My computations for this discrete version yield $\approx 2.97$, in fact, slightly larger than $r=3 \sqrt{3/\pi} \approx 2.93$ for the circle. Perhaps I miscalculated...