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Imagine there are two unrotated ellipses in 2d with different major and minor axes (that is to say different ellipses, but also consider case where ellipses have proportional major and minor axes, so same ellipses just that one is bigger than the other one) centered at origin. Should the shortest distance from one ellipse to the other be in the direction of either major or minor axes, or is it something completely else?

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In general, the shortest distance will not be in direction of one of the axes. Consider the extreme case of one ellipse degenerated to the line form $(-1,0)$ to $(+1,0)$ and the other going through $(\pm 2,0)$ and $(0,\pm1)$. If the shortest distance were along the axes, it would be 1, but you can find many shorter distances (in fact, here 1 is the longest distance in some sense).

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