In this work, we introduce a novel implicit representation of shape which is based on assigning to each pixel a probability that this pixel is inside the shape. This probabilistic representation of shape resolves two important drawbacks of alternative implicit shape representations such as the level set method: Firstly, the space of shapes is convex in the sense that arbitrary convex combinations of a set of shapes again correspond to a valid shape. Secondly, we prove that the introduction of shape priors into variational image segmentation leads to functionals which are convex with respect to shape deformations. For a large class of commonly considered (spatially continuous) functionals, we prove that - under mild regularity assumptions - segmentation and tracking with statistical shape priors can be performed in a globally optimal manner. In experiments on tracking a walking person through a cluttered scene we demonstrate the advantage of global versus local optimality.