This study investigates novel object-interaction priors for graph cut image segmentation with application to intervertebral disc delineation in magnetic resonance (MR) lumbar spine images. The algorithm optimizes an original cost function which constrains the solution with learned prior knowledge about the geometric interactions between different objects in the image. Based on a global measure of similarity between distributions, the proposed priors are intrinsically invariant with respect to translation and rotation. We further introduce a scale variable from which we derive an original fixed-point equation (FPE), thereby achieving scale-invariance with only few fast computations. The proposed priors relax the need of costly pose estimation (or registration) procedures and large training sets (we used a single subject for training), and can tolerate shape deformations, unlike template-based priors. Our formulation leads to an NP-hard problem which does not afford a form directly amenable to graph cut optimization. We proceeded to a relaxation of the problem via an auxiliary function, thereby obtaining a nearly real-time solution with few graph cuts. Quantitative evaluations over 60 intervertebral discs acquired from 10 subjects demonstrated that the proposed algorithm yields a high correlation with independent manual segmentations by an expert. We further demonstrate experimentally the invariance of the proposed geometric attributes. This supports the fact that a single subject is sufficient for training our algorithm, and confirms the relevance of the proposed priors to disc segmentation.