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By using the concept of gauge function, a pseudodistance function is defined for quantifying the clearance or the penetration depth of two convex point sets, depending on whether they separate or intersect. The linear programming formulations (for convex polyhedra) and the nonlinear constrained optimization formulations (for general convex objects) are presented for its calculation. For a pair of convex polyhedra, the pseudodistance function is differentiable almost everywhere with respect to the coordinate vectors of their vertices. Sufficient conditions for the differentiability and the characterization of its derivative are presented. By applying the pseudodistance function to the wrench space, a numerical measure of multifingered grasps is defined, which can be used for qualitative test and quantitative analysis of the force-closure property. On this basis, two algorithms for planning optimal force-closure grasps on general three-dimensional objects are developed. In addition, the application of the pseudodistance function in robot path planning is also demonstrated.