Dimension of the boundary in different metrics

TL;DR

This paper investigates how different metric choices, induced by continuous densities on Euclidean domains, affect the Hausdorff and packing dimensions of the domain boundaries.

Contribution

It provides a detailed analysis of boundary dimensions in various metrics induced by continuous densities on Euclidean domains.

Findings

Boundary dimensions vary with the choice of metric induced by densities

Hausdorff and packing dimensions are explicitly characterized

Results apply to a broad class of Euclidean domains

Abstract

We consider metrics on Euclidean domains $\Omega\subset\R^n$ that are induced by continuous densities $\rho\colon\Omega\rightarrow(0,\infty)$ and study the Hausdorff and packing dimensions of the boundary of $\Omega$ with respect to these metrics.