Oceanic surface flows are dominated by finite-time Lagrangian coherent structures that separate regions of qualitatively different dynamical behavior. Among these, eddy boundaries are of particular interest. Their exact identification is crucial for the study of oceanic transport processes and the investigation of impacts on marine life and the climate. Here, we present a novel method purely based on convexity, a condition that is intuitive and well-established, yet not fully explored. We discuss the underlying theory, derive an algorithm that yields comprehensible results and illustrate the presented method by identifying coherent structures and filaments in simulations and real oceanic velocity fields.