“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)
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.
