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arXiv · 2509.10943

Minimizing measures for the doubling condition

Fernando Benito F. de la Cigoña·José M. Conde Alonso·Pedro Tradacete
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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 $d\geq3$ 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.

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BibTeXRIS

Fernando Benito F. de la Cigoña, José M. Conde Alonso, Pedro Tradacete. 2025-09-13. Minimizing measures for the doubling condition. https://arxiv.org/abs/2509.10943

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