This work extends popular star-convexity and other more general forms of convexity priors. We represent an object as a union of "convex" overlappable subsets. Since an arbitrary shape can always be divided into convex parts, our regularization model restricts the number of such parts. Previous k-part shape priors are limited to disjoint parts. For example, one approach segments an object via optimizing its k coverage by disjoint convex parts, which we show is highly sensitive to local minima. In contrast, our shape model allows the convex parts to overlap, which both relaxes and simplifies the coverage problem, e.g. fewer parts are needed to represent any object. As shown in the paper, for many forms of convexity our regularization model is significantly more descriptive for any given k. Our shape prior is useful in practice, e.g. biomedical applications, and its optimization is robust to local minima.