Journal of Convex Analysis 32 (2025), No. 2, 585--603
Copyright Heldermann Verlag 2025
Conical Points of Convex Surfaces: Generalized Unit Normal and Angle Defect
Domenico Mucci
Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Università di Parma, Italy
domenico.mucci@unipr.it
Alberto Saracco
Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Università di Parma, Italy
alberto.saracco@unipr.it
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.
Keywords: Convex surfaces, conical points, unit normal, angle defect.
MSC: 26B25, 53A35, 26A45, 49J45.
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