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Hedgehogs are a natural generalization of convex bodies of class C+2. After recalling some basic facts concerning this generalization, we use the notion of index to study differential and integral geometries of hedgehogs.As applications, we prove a particular case of the Tennis Ball Theorem and a property of normals to a plane convex body of constant width.
We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.
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