# 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 $E_x$ that touches $x$ of second-order, the osculating ellipsoid at $x$. Denote the family of osculating ellipsoids by $F:=\{E_x:x\in\mbox{boundary of}~K\}$. Moreover, set $G:=\{TE: E\in F~\&~T\in SL(n)\}$. Is it true that there exists $\mathcal{E}\in G$ such that $\mathcal{E}\subseteq K.$

• If by 'inside', you are including the boundary of $K$, then, sure. Just arrange that $K$ actually agree with an ellipsoid in some closed set of directions and be outside the ellipsoid in the (open) complement. Presumably, though, you want it to be strictly inside except at the (two) points of contact (remember, you wanted 'origin-symmetric). However, even this can be arranged. Just have the closed set mentioned above consist of just two (opposite) points. – Robert Bryant Oct 2 '15 at 16:13
• I am not sure if I understood correctly, are you constructing $K$? Since I'm asking if for a given origin-symmetric body I can find an ellipsoid that is enclosed by the body. – K. P Oct 2 '15 at 16:26
• Yes, I'm constructing K, as I read your question as asking "Can it ever happen that one of these ellipsoids lies inside K?". If you meant to ask "Given K, does it always happen that at least one of these ellipsoids lies inside K?", then maybe you should do that. – Robert Bryant Oct 2 '15 at 17:07
• Sorry, I fixed it. – K. P Oct 2 '15 at 17:11
• What do you mean by «origin symmetric», and what is $n$ (I can make guesses, but it should be made explicit). – Loïc Teyssier Oct 3 '15 at 0:47