International Journal of Neutrosophic Science (IJNS) Vol. 24, No. 02, PP. 163-175, 2024 Foundations of neutrosophic convex structures Jos´ e Sanabria 1 ∗ , Ennis Rosas 2 , Elvis Aponte 3 1 Department of Mathematics, Faculty of Education and Sciences, University of Sucre, Sincelejo, Colombia 2 Department of Natural and Exact Sciences, Universidad de la Costa, Barranquilla, Colombia 3 Department of Mathematics, Faculty of Natural Sciences and Mathematics, Escuela Superior Polit´ ecnica del Litoral (ESPOL), Campus Gustavo Galindo km. 30.5 V´ ıa Perimetral, Guayaquil, Ecuador Emails: jesanabri@gmail.com; ennisrafael@gmail.com; ecaponte@espol.edu.ec Abstract In this paper an idea of neutrosophic convex structures (briefly, NC-structures) is given and some of their properties are explored. Also, NC-sets, neutrosophic concave sets and neutrosophic convex hull are defined and their properties are investigated. Moreover, the notions of NC-derived operator and NC-base are studied and their relationship to NC-structures are established. Keywords: Neutrosophic set; NC-space; neutrosophic hull operator; NC-derived operator; NC-base 1 introduction Convexity serves as a crucial and foundational characteristic across numerous branches of mathematics. But, some specific mathematical environments, such as vector spaces, are not the most suitable for studying the basic properties of convex sets. To avoid this problem, Van de Vel 17 introduced abstract convex structures (in short, convex structures) in terms of three axioms similar to those used to define topologies. Nowadays, convexity theory has become a branch of mathematics that deals with set-theoretic structures satisfying axioms similar to the usual convex sets. In fact, convex structures have appeared in research areas such as lattices, 16 graphs, 15 and topology. 18 Concretely, a collection C of subsets of a set X is a convex structure over X, if ∅ and X belong to C , moreover C is closed under arbitrary intersections and is closed under unions of chains. In this case, the pair (X, C ) is said to be a convex space and the members of C are called convex sets. A convex structure is totally determined by a special type of operator that is analogous to the closure operator in topology, and is termed the convex hull operator, which is defined as the intersection of all convex sets containing a given subset of a convex space. Convex hull operators have been investigated not only from the generalized point of view of convex structures, but also in particular cases for finite point sets, simple polygons, Brownian motion, spatial curves and epigraphs of functions. Convex hull operators have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling and ethology. Because convex structures are defined similarly to topologies, the natural question arises that whether they can be characterized by derivative operators. To answer this question, Chen and Shen 5 introduced the notion of derivative operators on convex spaces, termed c-derivative operators, which are suitable in the study of both convex and antimatroid spaces. On the other hand, neutrosophic set theory was established by Smarandache 12 in 1999 as a mathematical tool that generalizes the notions of fuzzy set and intuitionistic fuzzy set, being the best choice in situations where fuzzy set and fuzzy logic cannot express false membership information and intuitionistic fuzzy set and in- tuitionistic fuzzy logic are not able to handle information indeterminacy. While fuzzy set theory states that https://doi.org/10.54216/IJNS.240214 Received: October 22, 2023 Revised: February 09, 2024 Accepted: April 20, 2024 163