In this work, we present a new method for convex shape representation, which is regardless of the dimension of the concerned objects, using level-set approaches. To the best of our knowledge, the proposed prior is the first one which can work for high dimensional objects. Convexity prior is very useful for object completion in computer vision. It is a very challenging task to represent high dimensional convex objects. In this paper, we first prove that the convexity of the considered object is equivalent to the convexity of the associated signed distance function. Then, the second order condition of convex functions is used to characterize the shape convexity equivalently. We apply this new method to two applications: object segmentation with convexity prior and convex hull problem (especially with outliers). For both applications, the involved problems can be written as a general optimization problem with three constraints. An algorithm based on the alternating direction method of multipliers is presented for the optimization problem. Numerical experiments are conducted to verify the effectiveness of the proposed representation method and algorithm.