This paper proposes concentrated geo-privacy (CGP), a privacy notion that can be considered as the counterpart of concentrated differential privacy (CDP) for geometric data. Compared with the previous notion of geo-privacy [ABCP13, CABP13], which is the counterpart of standard differential privacy, CGP offers many benefits including simplicity of the mechanism, lower noise scale in high dimensions, and better composability known as advanced composition. The last one is the most important, as it allows us to design complex mechanisms using smaller building blocks while achieving better utilities. To complement this result, we show that the previous notion of geo-privacy inherently does not admit advanced composition even using its approximate version. Next, we study three problems on private geometric data: the identity query, k nearest neighbors, and convex hulls. While the first problem has been previously studied, we give the first mechanisms for the latter two under geo-privacy. For all three problems, composability is essential in obtaining good utility guarantees on the privatized query answer.