Dexterous manipulation of objects by a robot system benefits from the ability to generate precise grasps. In this paper, we propose the concept of Fingertip Space and its use for synthesizing precision grasps. Fingertip Space is a representation that takes into account both the local geometry of the object surface and the fingertip geometry. As such, it is directly applicable to the object point cloud data and establishes a basis for the grasp search space. We propose a model for hierarchically encoding the Fingertip Space that enables multilevel refinement for efficient grasp synthesis. The proposed method works at the grasp contact level while not neglecting object shape or hand kinematics. Experimental evaluation is performed for the Barrett hand, considering noisy and incomplete point cloud data.