Tight sign-central matrices

A real matrix A is called sign-central if the convex hull of the columns of contains the zero vector 0 for every matrix with the same sign pattern as A. A sign-central matrix A is called a minimal sign-central matrix if the deletion of any of the columns of A breaks the sign-centrality of A. A sign-central matrix A is called tight sign-central if the Hadamard (entrywise) product of any two columns of A contains a negative component. In this paper, we show that every tight sign-central matrix is minimal sign-central and characterize the tight sign-central matrices. We also determine the lower bound of the number of columns of a tight sign-central matrix in terms of the number of rows and the number of zero entries of the matrix.