Year: 2025
Type: preprint
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 complete 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 $d\geq3$ there are infinitely many independent minimizers when considering the Euclidean metric. 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agencia Estatal de Investigación
OpenAlex ID: W4415087508
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