Conical Points of Convex Surfaces: Generalized Unit Normal and Angle Defect

Given a conical point of a convex surface, a robust notion of generalized unit normal is introduced. Its relationship with the polar to the tangent cone implies the BV regularity of our unit normal. We then show that the area of the geodesically convex hull of its image, called angle defect, describes the energy concentration of the Gauss curvature of smooth surfaces approximating the tangent cone at the conical point.