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 $\varepsilon$?