We introduce the concept of Local Force Closure. We define a local force closure grasp as a grasp which is capable of resisting some given external wrench as well as (through local variation in contact wrenches) any wrench in some neighborhood of the given wrench, with grasp quality exceeding some given threshold. Local force closure is useful in applications where a grasp only needs to resist some given external wrench, rather than fully constraining object, but where there is some uncertainty regarding the exact external wrench that needs to be resisted, or where there is a possibility of having to cope with some (relatively small) unknown disturbance forces. We show that by allowing disc-shaped fingers in contact with convex vertices of a polygonal object, any given wrench can be resisted by just two frictionless fingers. For a given polygonal object with <i>n</i> vertices and an external wrench <i>w</i>_ext, we show how to find all pairs of features of <i>P</i>, that admit grasps capable of resisting <i>w</i>_ext with grasp quality greater or equal to some threshold <i>Q</i>, in <i>O(n^3/2+&#949;+K)</i> time, where <i>K</i> is the number of pairs in the output and <i>&#949;</i> is some arbitrarily small, positive constant. We then show how to adapt our algorithm to guarantee that the features reported, admit local force closure grasps.