Proposition: Fixed comparison errors do not transfer regularity

A bounded multiplicative comparison with a smooth strictly convex body does not imply that the compared body is smooth or strictly convex. Even arbitrarily small Hausdorff error does not imply either property for the approximating bodies.

Proof. Inscribed regular polygons converge in Hausdorff distance to the Euclidean disk. Each polygon has flat edges and corners. For sufficiently many sides it lies between $(1-\epsilon )$ and $(1+\epsilon )$ dilates of the disk. Thus neither conclusion follows from a fixed comparison allowance.