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Consider the hyperellipsoid $A$ in $\mathbb{R}^d$ given by the semi-major axes $a_1,a_2,\ldots,a_d$. Do points on the surface of the hyperellipsoid $A'$ with semi-major axes $a_1-\varepsilon, a_2-\varepsilon,\ldots,a_d-\varepsilon$ all have distance $\varepsilon$ to the original ellipsoid $A$? (assuming $a_i>\varepsilon$ for $i=1,\ldots,d$)

If not, how good of an approximation is this for $a_i>>\varepsilon$ in relation to $\varepsilon$?

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    after computing the distance in the 2d case by just inserting into the formula of an ellipse this is apparently not true. But how could an expression for the error (distance of point on $A'$ to point exactly $\varepsilon$ from A) be derived in the general case ($d$ dimensions)?2012-11-19
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    Related questions in 2D: [Uniform thickness border around skewed ellipse?](http://math.stackexchange.com/q/30219/856), [Does using an ellipse as a template still produce an ellipse?](http://math.stackexchange.com/q/64840/856)2012-11-19

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