The average number of normals through a point in a convex body and a related Euler-type identity

Anormal to a convex bodyK at a pointx in the boundary ofK is a ray with endpoint atx, perpendicular to support planeH ofK atx, and contained in the halfspace bounded byH that containsK. This work establishes bounds on the average number ofnormals through a point inK forK a polytope and forK with sufficiently smooth boundary. The integrals used to establish these bounds lead to an Euler-type relation.