It is stated in the book Convex Optimization, Boyd in page 47 that the ellipsoid 2 is the minimal because no other ellipsoid (centered at the origin) contains the point (top point) and is contained in Ellipsoid 2. However, I just draw an ellipsoid (red color) inside the ellipsoid 2. So, by which proof it is said that there is no ellipsoid inside ellipsoid 2 centered at origin passing through the top point?
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The minimal ellipsoid (with respect to area) centered at the lower dot and containing the upper dot in your figure is the degenerate ellipsoid with smaller semiaxis $=0$ and one apex at the upper dot; in short: a segment covered twice. The ${\cal E}_2$ in your figure has no minimal properties whatsoever.