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

TL;DR

This paper establishes a new lower bound on the number of convex components in compact sets with interior in ^n, generalizing previous inequalities and advancing understanding of convex decomposition in higher dimensions.

Contribution

It provides a generalized and improved lower bound on convex components of compact sets in ^n, extending prior results in the field.

Findings

Derived a new lower bound for convex components in ^n

Generalized previous inequalities in convex component analysis

Enhanced understanding of convex decomposition in higher dimensions

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.