Geodesic
Isoperimetric Inequalities in Normed Planes
Journal of convex analysis, Tome 29 (2022) no. 2, pp. 321-331
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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.
Classification : 52A10, 52A40
Mots-clés : Minkowski geometry, curves of constant width, Wigner caustic, isoperimetrix 
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     author = {R. S. dos Santos and M. Craizer},
     title = {Isoperimetric {Inequalities} in {Normed} {Planes}},
     journal = {Journal of convex analysis},
     pages = {321--331},
     year = {2022},
     volume = {29},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2022_29_2_JCA_2022_29_2_a0/}
}
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R. S. dos Santos; M. Craizer. Isoperimetric Inequalities in Normed Planes. Journal of convex analysis, Tome 29 (2022) no. 2, pp. 321-331. http://geodesic.mathdoc.fr/item/JCA_2022_29_2_JCA_2022_29_2_a0/