In this paper a new algorithm for segmentation of the foveal avascular zone in optical coherence tomography angiography images of the superficial capillary plexus is presented and evaluated. The algorithm is based on convolutional techniques, and for evaluation it has been compared with a collection of manual segmentations. Besides its performance, the main novelty presented is the ability to distinguish the purely avascular zone from the transitional environment whose importance has been recently pointed out. Its capability has been tested on images of patients with different types of diabetes mellitus, obtaining error rates between 1% and 1.5%. In addition, statistical data is shown for the segmented areas (including the transition zone, which had never been studied before) as a function of the type of diabetes. Moreover, a linear trend in outer and inner axis ratios is also observed. Overall, the algorithm represents a new approach in the analysis of optical coherence tomography angiography images, offering clinicians a new and reliable tool for objective foveal avascular zone segmentation of the superficial capillary plexus. Both the code and the dataset used are also made public in the cited repositories.