Remarks on the equality case of the Bonnesen inequality (Q453171)

Bonnesen's inequality for parallel sections of two convex bodies \(A,B\) is a refinement of the Brunn-Minkowski inequality. The authors give an elegant characterization of the equality case of the Bonnesen inequality using an operation called stretching. The \(2\)-dimensional case of this characterization has been settled before by \textit{G. A. Freiman} et al. [``Inverse additive problems for Minkowski sumsets. I'', Preprint, \url{arXiv:1105.5153}; J. Geom. Anal. 23, No. 1, 395--414 (2013; Zbl 1273.52004)]. The article contains also a proof of the equality case in a second inequality of Bonnesen dealing with projections. This result has been proved before by Freiman et al. and is a consequence of the first characterization.