Dimensions, Whitney covers, and tubular neighborhoods

TL;DR

This paper explores the relationships between dimensions, Whitney covers, and tubular neighborhoods in doubling metric spaces, providing characterizations of Minkowski and spherical dimensions through Whitney ball counts and analyzing boundary surface areas.

Contribution

It introduces new characterizations of Minkowski and spherical dimensions using Whitney ball counts in doubling metric spaces, linking geometric properties with metric dimension concepts.

Findings

Minkowski and spherical dimensions characterized via Whitney ball counts.

Connections established between dimensions and boundary surface area behavior.

Results applicable to Euclidean and general doubling metric spaces.

Abstract

Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface area of the boundaries of parallel sets. In particular, we give characterizations for the Minkowski and the spherical dimensions by means of the Whitney ball count.