Dimension, entropy, and the local distribution of measures

TL;DR

This paper introduces a unified approach using local entropy averages to analyze the local distribution of measures in Euclidean spaces, simplifying and extending existing results on homogeneity, porosity, and densities.

Contribution

It presents a general framework for studying local measure distributions, unifying and generalizing recent findings through entropy-based methods.

Findings

Unified approach to local measure distribution analysis

Generalized results on local homogeneity and porosity

Simplified proofs of conical density properties

Abstract

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local homogeneity, porosity and conical densities of measures.