Osculating ellipsoids

Let K be a given smooth, origin-symmetric, strictly convex body in n dimensional euclidean space. At every point x on the boundary of K there exists an origin-symmetric ellipsoid Ex that touches x of second-order, the osculating ellipsoid at x. Denote the family of osculating ellipsoids by F:={Ex:xboundary of K}. Moreover, set G:={TE:EF & TSL(n)}. Is it true that there exists EG such that EK.

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