Geometry of the Unit Sphere in Polynomial Spaces
Description
This brief presents a global perspective on the geometry of spaces of polynomials. Its particular focus is on polynomial spaces of dimension 3, providing, in that case, a graphical representation of the unit ball. Also, the extreme points in the unit ball of several polynomial spaces are characterized. Finally, a number of applications to obtain sharp classical polynomial inequalities are presented.
The study performed is the first ever complete account on the geometry of the unit ball of polynomial spaces. Nowadays there are hundreds of research papers on this topic and our work gathers the state of the art of the main and/or relevant results up to now. This book is intended for a broad audience, including undergraduate and graduate students, junior and senior researchers and it also serves as a source book for consultation. In addition to that, we made this work visually attractive by including in it over 50 original figures in order to help in the understanding of allthe results and techniques included in the book.
Reviews / Votes
"This book is the pictorial description of the unit sphere in 3-dimensional cases, which motivates analytic build-up in higher dimensions. This facilitates the reader's perception of the extreme point analysis of the polynomial spaces from a visual perspective. The extreme points of the polynomial spaces are crucial in studying polynomial inequalities. The classification of extreme points in polynomial spaces has a rich background in the literature. This book culminates in state-of-the-art results in this field of research." (Saikat Roy, Mathematical Reviews, June, 2025)
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Domingo García is a professor of mathematics at the University of Valencia, Spain. He spent two sabbatical years at Universidade Estadual de Campinas, Brazil, in 1993 and Kent State University, USA, in the academic year 2004-05, respectively. He has an extensive research related to Banach space theory, specifically in norm attaining operators, Dirichlet series, and holomorphic functions. He has been both plenary and invited speakers at several international conferences and is the author of more than 100 publications, including one book.
Manuel Maestre got his Ph.D. at the Universidad de Valencia (Spain) in 1982. He is corresponding academician of the Academy of Sciences of Madrid and editor of journals of mathematics. His main interests include functional analysis, and complex analysis in finite and infinite dimensions. He is the author of over 120 scientific publications, including several books. Currently he is a professor of mathematics at Universidad de Valencia (Spain).
Gustavo A. Mun~oz received his Ph.D. in Mathematics in 1999 from the Complutense University of Madrid (Spain), where he currently works as a member of the Department of Mathematical Analysis and Applied Mathematics. His research interests include geometry of Banach spaces of polynomials, polynomial inequalities, algebraic genericity (lineability and spaceability) and complexifications. In the last few years Prof. Muñoz has collaborated in several projects of mathematical modeling including two studies on the evolution of the pandemic of COVID-19.
Daniel L. Rodríguez is a Ph.D. Student at Universidad Complutense de Madrid (Spain) under the supervision of Profs. Gustavo A. Muñoz and Juan B. Seoane from Universidad Complutense de Madrid, and Krzysztof C. Ciesielski from West Virginia University (Morgantown, West Virginia, USA). His research interests include lineability, geometry of Banach spaces of polynomials, inequalities of polynomials, real analysis, set theory and mathematical foundations.
Juan B. Seoane received his first Ph.D. at the Universidad de Cádiz (Spain) jointly with Universität Karlsruhe (Germany) in 2005. His second Ph.D. was earned at Kent State University (Kent, Ohio, USA) in 2006. His main interests include functional analysis, set theory, mathematical foundations, and lineability. He is the author of over 200 scientific publications, including several books. Currently he is a professor at Universidad Complutense de Madrid (Spain) and editor of several mathematics journals.
