Minimizing measures for the doubling condition

2025/09/13 by Fernando Benito F. de la Cigoña, de la Cigoña, Fernando Benito F., José M. Conde Alonso +3
Engineering · #28A75 #30L99 #31C05 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.10943

openalex publication_date 2025/09/13 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We study those measures whose doubling constant is the least possible among doubling measures on a given metric space. It is shown that such measures exist on every metric space supporting at least one doubling measure. In addition, a connection between minimizers for the doubling constant and superharmonic functions is exhibited. This allows us to show that for the particular case of the euclidean space \mathbb R^d, Lebesgue measure is the only minimizer for the doubling constant (up to constant multiples) precisely when d=1 or d=2, while for d≥3 there are infinitely many independent minimizers. Analogously, in the discrete setting, we can show uniqueness of the counting measure as a minimizer for regular graphs where the standard random walk is a recurrent Markov chain. The counting measure is also shown to be a minimizer in every infinite graph where the cardinality of balls depends solely on their radii.

Citations

• Doubling constants and spectral theory on graphs
• The least doubling constant of a path graph
• The least doubling constant of a metric measure space
• A doubling measure on \R^d can charge a rectifiable curve

Related

• The least doubling constant of a metric measure space
• Doubling constants and spectral theory on graphs
• The least doubling constant of a path graph
• On the existence of doubling measures with certain regularity properties
• The doubling metric and doubling measures