Uniformity and Loewner Condtions of Metric Spaces

In this paper, we establish four equivalent conditions in the metric space setting. These conditions concern the (inner) uniformity of metric spaces, and (locally) Loewner conditions, weak slice conditions and k-cap conditions of metric measure spaces. This investigation completes the related study started by Bonk, Heinonen, and Koskela in 2001. Also, two examples are constructed to demonstrate that the assumption of the regularity of the underlying spaces can not be removed.