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Let be closed, bounded, convex sets. The monotonicity of the surface areas tells us that
We give quantitative estimates from below of the difference in the cases and . As an application, considered a decomposition of a closed and bounded set into a number of convex pieces, we deduce an estimate from below of the minimal number of convex components that may exist.
Cite this article
Flavia Giannetti, Sharp geometric quantitative estimates. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 1–6DOI 10.4171/RLM/748