Most current deep learning based approaches for image segmentation require annotations of large datasets, which limits their application in clinical practice. We observe a mismatch between the voxelwise ground-truth that is required to optimize an objective at a voxel level and the commonly used, less time-consuming clinical annotations seeking to characterize the most important information about the patient (diameters, counts, etc.). In this study, we propose to bridge this gap for the case of multiple nested star-shaped objects (e.g., a blood vessel lumen and its outer wall) by optimizing a deep learning model based on diameter annotations. This is achieved by extracting in a differentiable manner the boundary points of the objects at training time, and by using this extraction during the backpropagation. We evaluate the proposed approach on segmentation of the carotid artery lumen and wall from multisequence MR images, thus reducing the annotation burden to only four annotated landmarks required to measure the diameters in the direction of the vessel's maximum narrowing. Our experiments show that training based on diameter annotations produces state-of-the-art weakly supervised segmentations and performs reasonably compared to full supervision. We made our code publicly available at https://gitlab.com/radiology/aim/carotid-artery-image-analysis/nested-star-shaped-objects.