Read Clarkson's survey about dimensions in metric spaces.

Read Kunal Talwar's Bypassing The Embedding

Study About Embedding

Revise Hierarchical Routing in Doubling Metrics

Read Bartal and Fackarenpoel decomposition

Doubling Metric is indeed a generalization of Euclidean metric. Read Gupta Krauthgamer Lee paper on bounded geometries.

Work of Indyk, Linial and Matousek on applications of metrics in algorithms

Indyk and Matousek's Chapter 8