JournalsrlmVol. 28, No. 1pp. 1–6

Sharp geometric quantitative estimates

  • Flavia Giannetti

    Università di Napoli Federico II, Italy
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Abstract

Let EBRnE \subset B \subset \mathbb R^n be closed, bounded, convex sets. The monotonicity of the surface areas tells us that

Hn1(E)Hn1(B).\mathcal{H}^{n-1}(\partial E) \leqslant \mathcal{H}^{n-1}(\partial B).

We give quantitative estimates from below of the difference δ(E;B)=Hn1(B)Hn1(E)\delta(E;B)=\mathcal{H}^{n-1}(\partial B)-\mathcal{H}^{n-1}(\partial E) in the cases n=2n=2 and n=3n=3. As an application, considered a decomposition of a closed and bounded set into a number kk 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. 1–6

DOI 10.4171/RLM/748