Calculus of Variations and Geometric Measure Theory

F. Giannetti - G. Stefani

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

created by stefani on 16 Jun 2021
modified on 07 Dec 2022


Accepted Paper

Inserted: 16 jun 2021
Last Updated: 7 dec 2022

Journal: Forum Math.
Year: 2021

ArXiv: 2106.08725 PDF


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