In this work, we discuss how to incorporate Gulshan's geodesic star convexity prior in a region-based approach for interactive image segmentation, called 'IFT segmentation by Seed Competition', which encompasses many popular methods, such as watersheds, and fuzzy connectedness. This convexity constraint eliminates undesirable intricate shapes, improving the segmentation of objects with more regular shape. We include a theoretical proof of the optimality of the new algorithm in terms of a global minimum of an energy function subject to the shape constraints. We also present an experimental evaluation that shows the obtained gains in accuracy for segmenting a variety of medical images, including MR images of the foot, CT thoracic studies of the liver, and MR images of the breast. © 2013 IEEE.