Smoothness of sets in Euclidean spaces

TL;DR

This paper investigates properties of smooth sets in Euclidean spaces, establishing a precise version of Hungerford's Theorem on boundary dimensions and demonstrating invariance under certain automorphisms.

Contribution

It provides a sharp form of Hungerford's Theorem and shows the invariance of smooth set properties under specific automorphisms in Euclidean spaces.

Findings

Proved a sharp form of Hungerford's Theorem on boundary Hausdorff dimension.

Established invariance of smooth set properties under a class of automorphisms.

Enhanced understanding of the geometric structure of smooth sets in Euclidean spaces.

Abstract

We study some properties of smooth sets in the sense defined by Hungerford. We prove a sharp form of Hungerford's Theorem on the Hausdorff dimension of their boundaries on Euclidean spaces and show the invariance of the definition under a class of automorphisms of the ambient space.