# Is the set of points of equal distance to the surface of an ellipsoid again an ellipsoid?

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)? – rngantner Nov 19 '12 at 12:16
– Rahul Nov 19 '12 at 18:21

I'm not aware of any specific known results about the quality of the approximation. Clearly it's pretty good when the $\epsilon$ values are small and very bad if any $\epsilon$ value is larger than the minimum radius of curvature of the original ellipse. In 2D, we have two curves with known (and fairly simple) equations, so figuring out the error distance between them shouldn't be too hard. I expect a few minutes with Mathematica would yield some insight.
I don't know anything about $d$ dimensions (unless $d$ happens to be 2 or 3). Offsetting problems are hard enough even in the low-dimension cases, so I've never gone beyond.