Four Non-linear Problems on Normed Spaces - Volume I

This book deals with the following open problems: The Lineability Problem for Functionals (Aron and Gurariy 2004); The Nowhere Density Problem for Functionals (Enflo 2005); The Minimum-Norm Problem for Translations (Aizpuru and García-Pacheco 2003); The Banach-Mazur Conjecture for Rotations (Banach and Mazur 1932). The first problem is on the linear structure of the set of norm-attaining functionals on a Banach space (it also considers the linear structure of the set of non-norm-attaining functionals). The second problem is on the topological structure of the set of norm-attaining functionals on a Banach space (it also considers the topological structure of the set of non-norm-attaining functionals). The third problem deals with the non-linear metric structure of translates of convex sets. Finally the last problem deals with the isometric structure of the unit ball of a Banach space.