Multicones, duality and invariance for families of matrices

TL;DR

This paper explores the geometric structure of multicones, generalizations of classical cones, and develops duality and invariance concepts to identify conditions for invariant multicones in matrix families.

Contribution

It introduces a new duality concept for multicones, analyzes their invariance properties, and provides conditions and a computational method for constructing invariant pairs.

Findings

Established sufficient conditions for invariant multicones based on eigenspace structures

Developed a practical procedure to construct invariant pairs of multicones

Extended classical cone duality to multicones with applications to matrix invariance

Abstract

In this paper we analyze the geometric structure and properties of a certain class of subsets of $\Bbb R^d$, known in the literature as 1-multicones, and here simply called multicones, which are quite natural generalizations of the classical cones. In particular, we introduce and investigate a suitable extension of the concept of duality to such multicones, which allows us to treat in a convenient way many issues related to their invariance and strict invariance for matrices and finite families of matrices. One of the main goals of our investigations is the detection of sufficient conditions on the structure of the eigenspaces of a given finite family of matrices to assure the existence of an embedded pair of invariant multicones, which are the smallest and the biggest in a suitable and natural sense. The conditions we find also suggest us a practical computational procedure for the actual construction of such invariant embedded pair.