Uniformly Metrizable Bornologies

This paper is devoted to study the following question: given a bornology $\mathbf{B}$ on a metric space (X,d), when is there a metric d' on X, uniformly equivalent to d, for which the bornology of d'-bounded sets coincides with $\mathbf{B}$? If that happens, we will say that $\mathbf{B}$ is uniformly metrizable. We characterize these kinds of bornologies on X, and we obtain necessary and sufficient conditions in order to some of the most common bornologies enjoy this property. More precisely, we will consider the compact bornology, the totally bounded bornology and the bornology of the Bourbaki-bounded sets.