Every mathematician working in Banaeh spaee geometry or Approximation theory knows, from his own experienee, that most "natural" geometrie properties may faH to hold in a generalnormed spaee unless the spaee is an inner produet spaee. To reeall the weIl known definitions, this means IIx 11 = *, where is an inner (or: scalar) product on E, Le. a function from ExE to the underlying (real or eomplex) field satisfying: (i) O for x o. (ii) is linear in x. (iii) = (intherealease, thisisjust =

The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordanvon Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications).

1. Introduction. 1.1. Historical notes. 1.2. Normed linear spaces. 1.3. Strictly convex normed linear spaces. 1.4. Inner product spaces. 1.5. Orthogonalities in normed linear spaces -- 2. Norm derivatives. 2.1. Norm derivatives : Definition and basic properties. 2.2. Orthogonality relations based on norm derivatives. 2.3. p'[symbol]-orthogonal transformations. 2.4. On the equivalence of two norm derivatives. 2.5. Norm derivatives and projections in normed linear spaces. 2.6. Norm derivatives and Lagrange's identity in normed linear spaces. 2.7. On some extensions of the norm derivatives. 2.8. p-orthogonal additivity -- 3. Norm derivatives and heights. 3.1. Definition and basic properties. 3.2. Characterizations of inner product spaces involving geometrical properties of a height in a triangle. 3.3. Height functions and classical orthogonalities. 3.4. A new orthogonality relation. 3.5. Orthocenters. 3.6. A characterization of inner product spaces involving an isosceles trapezoid property. 3.7. Functional equations of the height transform -- 4. Perpendicular bisectors in Normed spaces. 4.1. Definitions and basic properties. 4.2. A new orthogonality relation. 4.3. Relations between perpendicular bisectors and classical orthogonalities. 4.4. On the radius of the circumscribed circumference of a triangle. 4.5. Circumcenters in a triangle. 4.6. Euler line in real normed space. 4.7. Functional equation of the perpendicular bisector transform -- 5. Bisectrices in real Normed spaces. 5.1. Bisectrices in real normed spaces. 5.2. A new orthogonality relation. 5.3. Functional equation of the bisectrix transform. 5.4. Generalized bisectrices in strictly convex real normed spaces. 5.5. Incenters and generalized bisectrices -- 6. Areas of triangles in Normed spaces. 6.1. Definition of four areas of triangles. 6.2. Classical properties of the areas and characterizations of inner product spaces. 6.3. Equalities between different area functions. 6.4. The area orthogonality.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

In this volume, the contributing authors deal primarily with the interaction among problems of analysis and geometry in the context of inner product spaces. They present new and old characterizations of inner product spaces among normed linear spaces and the use of such spaces in various research problems of pure and applied mathematics. The methods employed are accessible to students familiar with normed linear spaces. Some of the theorems presented are at the same time simple and challenging.

The volume consists of a set of surveys on geometry in the broad sense. The goal is to present a certain number of research topics in a non-technical and appealing manner. The topics surveyed include spherical geometry, the geometry of finite-dimensional normed spaces, metric geometry (Bishop—Gromov type inequalities in Gromov-hyperbolic spaces), convexity theory and inequalities involving volumes and mixed volumes of convex bodies, 4-dimensional topology, Teichmüller spaces and mapping class groups actions, translation surfaces and their dynamics, and complex higher-dimensional geometry. Several chapters are based on lectures given by their authors to middle-advanced level students and young researchers. The whole book is intended to be an introduction to current research trends in geometry.