Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Convex surfaces
  • conical points
  • unit normal
  • angle defect

Mathematics Subject Classification

26B25, 53A35, 26A45, 49J45

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Suggested citation

D. Mucci, A. Saracco. “Conical Points of Convex Surfaces: Generalized Unit Normal and Angle Defect.” Journal of Convex Analysis 32 (2025), No. 2, 585–603. https://doi.org/10.68381/jca32029

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.