Computational geometry is a subfield of theoretical computer science devoted to the design andimplementation of geometric algorithms, as well as to their analysis, and to the combinatorialstructure that they manipulate. In particular, computational geometry encompasses a diversity ofoptimization problems. It is often infeasible in practice to find the optimal solution of a geometric optimization problem. This is because the optimum might typically be hard to compute. This suggests that instead of insisting on computing the exact solution for optimization problems, one should be satisfied with a possibly suboptimal solution that approximates the optimum reasonably well.The goal of this proposal is to consider optimization problems involving geometric objects in low dimensions, and design efficient algorithms that guarantee a good approximation for their solutions. Such problems have been well studied in abstract settings (that is, when the objects are abstract and no geometric properties are known), but their geometric variants have received much less attention, and the solutions to most of these problems have still remained elusive.This project suggests an interdisciplinary challenge. On the theoretical front, it aims to develop a set of mathematical tools taken from discrete geometry, such as geometric arrangements and epsilon nets, which exploit the geometric structure of the given setting. In fact, such tools can be exploited on a broader set of problems, and lie beyond the scope of the problems presented in this proposal. On the applied front, this problems have applications to other disciplines as sensor networking, computer graphics, geographic information systems, machine learning and more. In fact, the PI is collaborating with researchers from sensor networking and learning where she applies such mathematical tools in order to solve problems from the real world.