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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?

${}{}{}{}{}{}{}$

it is copied from the book: Convex Optimization, by Boyd

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  • $\begingroup$ In my version of the book, there is a third dot in this figure. Maybe that dot is missing in your version? $\endgroup$
    – gerw
    Commented Jun 5, 2016 at 17:46

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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.

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    $\begingroup$ From the caption of the figure it seems to me that "area" is not used to define "minimal ellipsoid", but "minimal" is to be understood in the "subset" sense. $\endgroup$
    – gerw
    Commented Jun 4, 2016 at 18:25

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