Given a polygon, perturb its vertices while maintaining a constraint that none of the edges intersect. Perhaps also keep all the vertices within a unit disc. The motivation was to continuously sample among all possible polygons of a given number of vertices.
In three dimensions, easy is if all the faces are triangles. If not triangles, then difficult seems to be, when a perturbation causes a face with 4 or more vertices to no longer be planar, to construct a minimal surface within a boundary of straight edges, then forbid those surface faces from intersecting.
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