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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.
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.