An error estimate for the isoperimetric deficit
The author introduces a new geometric tool -- cyclic rearrangement -- which dissects some planar star bodies into four parts and then glue them around a rhombus in such a way that the new set preserves much of the boundary structure of the original body (including its perimeter) while potentially increases the area. Some geometric properties of this new tool are analysed, and some applications are also obtained: 1) A new proof of the planar isoperimetric inequality. 2) New approaches to Bonnesen-style error estimates for the isoperimetric deficit of planar star bodies that are centrally symmetric with respect to the origin. The resulting error bounds do not give the strongest estimates obtained up to now by Fuglede, but they do improve the classical result of Bonnesen in several interesting cases and they simplify the proof substantially.
scientific article; zbMATH DE number 5015543