cvgmt: On the convex components of a set in $\mathbb{R}^n$

Accepted Paper

Inserted: 16 jun 2021
Last Updated: 7 dec 2022

Journal: Forum Math.
Year: 2021

Abstract:

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008–2009) 71–78.

Keywords: Hausdorff distance, convex body, convex component, monotonicity of perimeter

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On the convex components of a set in $\mathbb{R}^n$