Finding minimal width of polygon

Trying to find the minimal width of a polygon as described as any point on any line, connected to any other point on any other line without going outside the shape, or across a hole. This must work for Convex, Concave, and polygons with holes. The shapes are presented as bezier curves.

This is easy to do visually, but I can't imagine any kind of method that would be effeicient, as lines and curves contain an infinite number of points.