Contact points of the greatest ellipsoid in a convex body and an application to convex curves (Q1039859)

It is well known that the maximal-volume ellipsoid \(E\) in a convex body \(B\) is unique and has many points \(p\in B \cap E\) that fix it within \(B\). The author establishes some restrictions on location of such contact points which imply the following main result: For any convex body \(B\) in the Euclidean plane symmetric about the origin 0, there exists a linear transformation \(L\) such that \(L(B)\) contains the unit circle \(U\) centered at 0 and fits either into a square circumscribed to \(U\) or into a centrally symmetric hexagon with no acute angles also circumscribed to \(U\).