Segmenting an image into multiple components is a central task in computer vision. In many practical scenarios, prior knowledge about plausible components is available. Incorporating such prior knowledge into models and algorithms for image segmentation is highly desirable, yet can be non-trivial. In this work, we introduce a new approach that allows, for the first time, to constrain some or all components of a segmentation to have convex shapes. Specifically, we extend the Minimum Cost Multicut Problem by a class of constraints that enforce convexity. To solve instances of this NP-hard integer linear program to optimality, we separate the proposed constraints in the branch-and-cut loop of a state-of-the-art ILP solver. Results on photographs and micrographs demonstrate the effectiveness of the approach as well as its advantages over the state-of-the-art heuristic.