On the convex components of a set in ℝ n

Abstract We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n {\mathbb{R}^{n}} for all n ≥ 2 {n\geq 2} . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78].