Affine-regular hexagons of extreme areas inscribed in a centrally symmetric convex body

Let M be a planar centrally symmetric convex body. If H is an affine regular hexagon of the smallest (the largest) possible area inscribed in M, then M contains (respectively, the interior of M does not contain) an additional pair of symmetric vertices of the affine-regular 12-gon T-H whose every second vertex is a vertex of H. Moreover, we can inscribe in M an octagon whose three pairs of opposite vertices are vertices of an affine-regular hexagon H and the remaining pair is a pair of opposite vertices of T-H. A corollary concerns packing M with its three homothetical copies. Another corollary is that the unit disk of any Minkowski plane contains three points in distances at least 1 + root3/3.