Pyramidal Polytopes in the Stability Region

TL;DR

This paper introduces a new class of stable vectors called pyramidal polytopes, whose convex hulls are stable, providing insights into the geometric structure of stability regions for monic polynomials.

Contribution

It defines pyramidal polytopes of stable vectors and demonstrates that their convex hulls are also stable, advancing understanding of stability regions in polynomial analysis.

Findings

Defined pyramidal polytopes of stable vectors

Proved convex hulls of these polytopes are stable

Enhanced geometric understanding of stability regions

Abstract

Every $n th$ order monic polynomial corresponds $n$-dimensional vector. If the given polynomial is stable that is all its roots lie in the open left half plane it is said to be Hurwitz polynomial and the corresponding vector is called stable vector. The set of stable vectors is non-convex. In this paper, we define special $(n+1) $ stable vectors such that their convex hull is stable.