On intervals and sets of hypermatrices (tensors)

Interval hypermatrices (tensors) are introduced and interval \alpha-hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where \alpha can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E₀-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P₀)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.