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I have a set $X$ of points in $\mathbb{R}^2$ and I'm trying to find the smallest encompassing ellipse which main axes are parallel to the coordinate system's (to put it differently, its both centres share one coordinate). I need the gravitational centre and the vertical and horizontal radius.

Now, I managed to do this for a horizontally aligned rectangle, but that isn't much help (although it's a first approximation, as I can easily draw a rectangle around $X$). I also found some formulas on the internet, but they seem to be wrong as I keep getting $0$ for the radius.

Can this be done algebraically?

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    Minimal with respect to what? Minimal area, perimeter, $\max\{a,b\}$ or what?2012-10-05
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    @enzotib: Minimal area.2012-10-05

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