We consider a class of multi-agent optimal coverage problems in which the goal is to determine the optimal placement of a group of agents in a given mission space so that they maximize a coverage objective that represents a blend of individual and collaborative event detection capabilities. This class of problems is extremely challenging due to the non-convex nature of the mission space and of the coverage objective. With this motivation, greedy algorithms are often used as means of getting feasible coverage solutions efficiently. Even though such greedy solutions are suboptimal, the submodularity (diminishing returns) property of the coverage objective can be exploited to provide performance bound guarantees. Moreover, we show that improved performance bound guarantees (beyond the standard (1-1/e) performance bound) can be established using various curvature measures of the coverage problem. In particular, we provide a brief review of all existing popular applicable curvature measures, including a recent curvature measure that we proposed, and discuss their effectiveness and computational complexity, in the context of optimal coverage problems. We also propose novel computationally efficient techniques to estimate some curvature measures. Finally, we provide several numerical results to support our findings and propose several potential future research directions.