Tube formulas for self-similar fractals

TL;DR

This paper reviews recent advances in deriving explicit tube formulas for self-similar fractals, highlighting their applications in geometry, spectral theory, and fractal analysis.

Contribution

It introduces new developments in tube formulas specifically for self-similar fractals, connecting geometric properties with spectral and measure-theoretic aspects.

Findings

Explicit tube formulas for self-similar fractals are developed.

Applications include understanding fractal dimensions and curvature measures.

Connections to spectral asymptotics and geometric analysis are established.

Abstract

Tube formulas (by which we mean an explicit formula for the volume of an $\epsilon$-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl's celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, the tube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which are not too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here.