Journal of Convex Analysis 29 (2022), No. 2, 321--331
Copyright Heldermann Verlag 2022
Isoperimetric Inequalities in Normed Planes
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
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.
Keywords: Minkowski geometry, curves of constant width, Wigner caustic, isoperimetrix.
MSC: 52A10, 52A40.
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