Domingo García is Professor of mathematics at the University of Valencia, Spain, where he also is Head of Department. 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.Mingu Jung completed his Ph.D. in 2021 at Pohang University of Science and Technology (POSTECH) under the supervision of Yun Sung Choi and Manuel Maestre. His research focuses primarily on functional analysis and nonlinear analysis. He is currently a CMC Fellow (Hyo Chul Myung Assistant Professor) at the June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study (KIAS).Manuel Maestre got his Ph.D. at the Universidad de Valencia (Spain) in 1982. He is corresponding academician of the Academy of Sciences of Spain 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 more than 120 scientific publications,including several books. Currently he is a Professor of Mathematics at Universidad de Valencia (Spain).Gustavo A. Muñoz Fernández graduated in Mathematics and Physics and obtained a PhD in Mathematics from the Complutense University of Madrid (Spain) in 1999. His main mathematical interests lie in Functional Analysis, particularly in the study of polynomial spaces in normed spaces, the geometry of polynomial spaces, and polynomial inequalities. He has also conducted extensive research on algebraic genericity, specifically the lineability of spaces of rare functions. Currently, he is a Professor at the Complutense University of Madrid and the director of the Interdisciplinary Mathematics Institute (IMI).Juan B. Seoane Sepúlveda received his first Ph.D. at the Universidad de C´adiz (Spain) jointly with Universit¨at Karlsruhe (Germany) in 2005 under the supervision of Profs. Richard M. Aron and Antonio Aizpuru. His second Ph.D. was earned at Kent State University (Kent, Ohio, USA) in 2006 under the supervision of Profs. Richard M. Aron and Vladimir I. Gurariy. His main interests include Real and Complex Analysis, Operator Theory, Geometry of Banach spaces, History of Mathematics, Lineability and Spaceability, and Mathematical Biology. He is currently a Professor at Universidad Complutense de Madrid (Spain).

• Preface Author Biographies I Polynomials on Unbalanced Convex Bodies 1 Preliminaries II The Geometry of Homogeneous Trinomials on the Unit Square 2 Preliminaries III The Krein-Milman Approach, Classical Inequalities and Applications 3 Preliminaries BibliographyList of Figures Index

Domingo García is Professor of mathematics at the University of Valencia, Spain, where he also is Head of Department. 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.Mingu Jung completed his Ph.D. in 2021 at Pohang University of Science and Technology (POSTECH) under the supervision of Yun Sung Choi and Manuel Maestre. His research focuses primarily on functional analysis and nonlinear analysis. He is currently a CMC Fellow (Hyo Chul Myung Assistant Professor) at the June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study (KIAS).Manuel Maestre got his Ph.D. at the Universidad de Valencia (Spain) in 1982. He is corresponding academician of the Academy of Sciences of Spain 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 more than 120 scientific publications,including several books. Currently he is a Professor of Mathematics at Universidad de Valencia (Spain).Gustavo A. Muñoz Fernández graduated in Mathematics and Physics and obtained a PhD in Mathematics from the Complutense University of Madrid (Spain) in 1999. His main mathematical interests lie in Functional Analysis, particularly in the study of polynomial spaces in normed spaces, the geometry of polynomial spaces, and polynomial inequalities. He has also conducted extensive research on algebraic genericity, specifically the lineability of spaces of rare functions. Currently, he is a Professor at the Complutense University of Madrid and the director of the Interdisciplinary Mathematics Institute (IMI).Juan B. Seoane Sepúlveda received his first Ph.D. at the Universidad de C´adiz (Spain) jointly with Universit¨at Karlsruhe (Germany) in 2005 under the supervision of Profs. Richard M. Aron and Antonio Aizpuru. His second Ph.D. was earned at Kent State University (Kent, Ohio, USA) in 2006 under the supervision of Profs. Richard M. Aron and Vladimir I. Gurariy. His main interests include Real and Complex Analysis, Operator Theory, Geometry of Banach spaces, History of Mathematics, Lineability and Spaceability, and Mathematical Biology. He is currently a Professor at Universidad Complutense de Madrid (Spain).

• Preface Author Biographies I Polynomials on Unbalanced Convex Bodies 1 Preliminaries II The Geometry of Homogeneous Trinomials on the Unit Square 2 Preliminaries III The Krein-Milman Approach, Classical Inequalities and Applications 3 Preliminaries BibliographyList of Figures Index

Domingo García is Professor of mathematics at the University of Valencia, Spain, where he also is Head of Department. 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.Mingu Jung completed his Ph.D. in 2021 at Pohang University of Science and Technology (POSTECH) under the supervision of Yun Sung Choi and Manuel Maestre. His research focuses primarily on functional analysis and nonlinear analysis. He is currently a CMC Fellow (Hyo Chul Myung Assistant Professor) at the June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study (KIAS).Manuel Maestre got his Ph.D. at the Universidad de Valencia (Spain) in 1982. He is corresponding academician of the Academy of Sciences of Spain 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 more than 120 scientific publications,including several books. Currently he is a Professor of Mathematics at Universidad de Valencia (Spain).Gustavo A. Muñoz Fernández graduated in Mathematics and Physics and obtained a PhD in Mathematics from the Complutense University of Madrid (Spain) in 1999. His main mathematical interests lie in Functional Analysis, particularly in the study of polynomial spaces in normed spaces, the geometry of polynomial spaces, and polynomial inequalities. He has also conducted extensive research on algebraic genericity, specifically the lineability of spaces of rare functions. Currently, he is a Professor at the Complutense University of Madrid and the director of the Interdisciplinary Mathematics Institute (IMI).Juan B. Seoane Sepúlveda received his first Ph.D. at the Universidad de C´adiz (Spain) jointly with Universit¨at Karlsruhe (Germany) in 2005 under the supervision of Profs. Richard M. Aron and Antonio Aizpuru. His second Ph.D. was earned at Kent State University (Kent, Ohio, USA) in 2006 under the supervision of Profs. Richard M. Aron and Vladimir I. Gurariy. His main interests include Real and Complex Analysis, Operator Theory, Geometry of Banach spaces, History of Mathematics, Lineability and Spaceability, and Mathematical Biology. He is currently a Professor at Universidad Complutense de Madrid (Spain).

• Preface Author Biographies I Polynomials on Unbalanced Convex Bodies 1 Preliminaries II The Geometry of Homogeneous Trinomials on the Unit Square 2 Preliminaries III The Krein-Milman Approach, Classical Inequalities and Applications 3 Preliminaries BibliographyList of Figures Index