Smoothness of Minkowski sum and generic rotations

TL;DR

This paper investigates how the smoothness of the Minkowski sum of two convex bodies can be affected by rotation, showing conditions under which smoothness can or cannot be improved through generic rotations.

Contribution

It constructs examples of convex bodies where rotation does not improve smoothness and identifies conditions ensuring the Minkowski sum's smoothness after rotation.

Findings

Rotating one convex body may not increase the Minkowski sum's smoothness.

Certain curvature conditions guarantee smoothness after generic rotation.

Improves previous results on Minkowski sum smoothness.

Abstract

Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other hand, if for one of the bodies the zero set of the Gaussian curvature has countable spherical image, we show that any generic rotation makes their Minkowski sum as smooth as the summands. We also improve and clarify some previous results on smoothness of the Minkowski sum.