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This paper considers basket grasps, where a two-finger robot hand forms a basket that can safely lift and carry rigid objects in a 2-D gravitational environment. The two-finger basket grasps form special points in a high-dimensional configuration space of the object and two-finger robot hand. This paper establishes that all two-finger basket grasps can be found in a low-dimensional contact space that parametrizes the two-finger contacts along the supported object boundary. Using contact space, each basket grasp is associated with its depth that provides a security measure while carrying the object, as well as its safety margin away from a critical finger opening where the object drops-off into its intended destination. Geometric techniques that compute the depth and drop-off finger opening are described and illustrated with detailed graphical and numerical examples., QC 20210315