We provide for every 2 ≤ k ≤ n an n-dimensional Banach space E with a unique distance ellipsoid 𝓔 such that there are precisely k linearly independent contact points between 𝓔 and $B_E$. The corresponding result holds for spaces with non-unique distance ellipsoids as well. We construct n-dimensional Banach spaces E such that one distance ellipsoid has precisely k linearly independent contact points and all other distance ellipsoids have less than k-1 such points.
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