[math-fun] Condition for a twisted shape to stand on its own on a horizontal surface

Suppose we have a rigid surface X of uniform density in R^3, that can be any shape such that it is topologically a 2-disk, part of whose boundary is a curve C lying in a unique plane that does not intersect X anywhere else, and such that when this curve C is placed on a horizontal plane R^2, the shape stands up on its own, stably. What is the mathematical condition that X stand stably when C is placed on R^2 (the xy-plane), it stands stably on its own ? Assuming a vertical gravity. (With "stably" meaning that if X is tilted in any direction on R^2 by some small enough eps > 0, it will return to a standing position with C on R^2 again.) —Dan