Abstract

The classical isoperimetric inequality can be extended to a general normed plane (see H. Busemann: The isoperimetric problem in the Minkowski plane, Amer. J. Math. 69/4 (1947) 863–871). In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we consider normed planes with piecewise smooth unit balls and the corresponding class of admissible curves. For such an admissible curve, the singular sets are defined as projections in the subspaces of symmetric and constant width admissible curves. In this context, we obtain some improved isoperimetric inequalities whose equality hold for symmetric or constant width curves.

Author information

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

Rafael S. dos Santos

Department of Mathematics, Pontifical University, Rio de Janeiro, Brazil

rsegadas@gmail.com

Marcos Craizer

Department of Mathematics, Pontifical University, Rio de Janeiro, Brazil

craizer@puc-rio.br

Keywords

  • Minkowski geometry
  • curves of constant width
  • Wigner caustic
  • isoperimetrix

Mathematics Subject Classification

52A10, 52A40

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Suggested citation

R. S. dos Santos, M. Craizer. “Isoperimetric Inequalities in Normed Planes.” Journal of Convex Analysis 29 (2022), No. 2, 321–331. https://doi.org/10.68381/jca29019

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.