When all parallel sets of a $ C^1 $-hypersurface are nowhere $ C^1 $-regular

TL;DR

This paper characterizes when all parallel sets of a $ C^1 $-hypersurface lack $ C^1 $-regularity, revealing that for generic convex bodies, interior parallels also lack $ C^1 $-regularity.

Contribution

It provides a necessary and sufficient condition for $ C^1 $-hypersurfaces to have all parallel sets non-$ C^1 $-regular, advancing understanding of regularity properties of parallel bodies.

Findings

All interior parallel bodies of a generic $ C^1 $-regular convex body have nowhere $ C^1 $-regular boundaries.

A precise condition characterizes when a $ C^1 $-hypersurface's parallel sets are nowhere $ C^1 $-regular.

Abstract

We prove a necessary and sufficient condition for a $ C^1 $-hypersurface to have all parallel sets nowhere $ C^1 $-regular. As a corollary, we deduce that for a generic $ C^1 $-regular convex body all interior parallel bodies have nowhere $ C^1 $-regular boundaries.