Pestov–Ionin theorem

The Pestov–Ionin theorem in the differential geometry of plane curves states that every simple closed curve of curvature at most one encloses a unit disk.

Although a version of this was published for convex curves by Wilhelm Blaschke in 1916,[1] it is named for German Gavrilovich Pestov [ru] and Vladimir Kuzmich Ionin [ru], who published a version of this theorem in 1959 for non-convex doubly differentiable (

[2] The theorem has been generalized further, to curves of bounded average curvature (singly differentiable, and satisfying a Lipschitz condition on the derivative),[3] and to curves of bounded convex curvature (each point of the curve touches a unit disk that, within some small neighborhood of the point, remains interior to the curve).

[4] The theorem has been applied in algorithms for motion planning.

In particular it has been used for finding Dubins paths, shortest routes for vehicles that can move only in a forwards direction and that can turn left or right with a bounded turning radius.