The extreme points of certain polytopes of doubly substochastic matrices

Abstract

Let ${\omega }_n^{\pi }$ and ${\omega }_n^{t\mathrm{\&}h}$ denote the convex polytope of $n\times n$ centrosymmetric doubly substochastic matrices and the convex polytope of $n\times n$ symmetric and Hankel-symmetric doubly substochastic matrices, respectively. In this paper, we investigate and fully characterize the extreme points of ${\omega }_n^{\pi }$ and ${\omega }_n^{t\mathrm{\&}h}$ which generalizes the results by Brualdi and Cao in [Brualdi RA, Cao L. Symmetric, Hankel-symmetric, and centrosymmetric doubly stochastic matrices. ActaMath Vietnam. 2018;43:675–700].

COMMUNICATED BY:

• Ravindra B. Bapat

Keywords:

• Centrosymmetric matrices
• doubly substochastic matrices
• extreme points

AMS Subject Classifications:

• 05C50
• 15B51

Acknowledgments

We are grateful to the anonymous referees for their very valuable comments on our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Lei Cao  http://orcid.org/0000-0001-7613-7191

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (No. 11601233); the Fundamental Research Funds for the Central Universities (No. KJQN201718); the Natural Science Foundation of Jiangsu Province (BK20160708). The third author is supported by the National Natural Science Foundation of China (No. 11571220).