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• A computational binary characterization method for convex object(s). • A unified model for convex object(s) segmentation and convex hull computation. • Linearization technique employed to reduce the difficulty of the model solving. • A convergent proximal alternating direction method of multipliers for the linearization problem. • An interactive procedure to improve the segmentation accurac y in terms of shape distance and D ice coefficient. Convexity prior is one of the main cues for human vision and shape completion with important applications in image processing and computer vision. This paper provides a computable characterization method for shape convexity, and illustrates its applications in image segmentation and convex hull computation. We prove that the convexity of a region is equivalent to a series of quadratic inequality constraints on its indicator function. By incorporating this result, models are proposed for image segmentation with convexity prior and convex hull computation of clean and noisy data sets, respectively. Then, these models are summarized into a unified optimization problem on binary function(s) with the quadratic inequality constraints. Numerical method is proposed by linearizing the quadratic constraints. The linearization problem is solved by a proximal alternating direction method of multipliers, the convergence of which is guaranteed under some proper conditions. Numerical experiments demonstrate the efficiency and effectiveness of the proposed methods for image segmentation and convex hull computation. [ABSTRACT FROM AUTHOR]