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Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers. It should also be of use to people working in other areas of mathematics and in the applied fields.

Convex geometry --- Discrete geometry --- Géométrie convexe --- Géométrie discrète --- Convex geometry. --- Discrete geometry. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematical Theory --- 514.17 --- Combinatorial geometry --- Convex sets. Geometric figure arrangements. Geometric inequalities --- 514.17 Convex sets. Geometric figure arrangements. Geometric inequalities --- Mathematics. --- Convex and Discrete Geometry. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry .

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This book provides a comprehensive presentation of geometric results, primarily from the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. An important feature of the book is that all necessary tools from the classical theory of spherical harmonics are presented with full proofs. These tools are used to prove geometric inequalities, stability results, uniqueness results for projections and intersections by hyperplanes or half-spaces and characterisations of rotors in convex polytopes. Again, full proofs are given. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This treatise will be welcomed both as an introduction to the subject and as a reference book for pure and applied mathematics.

Convex sets. --- Fourier series. --- Spherical harmonics. --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Fourier analysis --- Sets, Convex --- Convex domains --- Set theory

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In this monograph the author presents the theory of duality for nonconvex approximation in normed linear spaces and nonconvex global optimization in locally convex spaces. Key topics include: * duality for worst approximation (i.e., the maximization of the distance of an element to a convex set) * duality for reverse convex best approximation (i.e., the minimization of the distance of an element to the complement of a convex set) * duality for convex maximization (i.e., the maximization of a convex function on a convex set) * duality for reverse convex minimization (i.e., the minimization of a convex function on the complement of a convex set) * duality for d.c. optimization (i.e., optimization problems involving differences of convex functions). Detailed proofs of results are given, along with varied illustrations. While many of the results have been published in mathematical journals, this is the first time these results appear in book form. In addition, unpublished results and new proofs are provided. This monograph should be of great interest to experts in this and related fields. Ivan Singer is a Research Professor at the Simion Stoilow Institute of Mathematics in Bucharest, and a Member of the Romanian Academy. He is one of the pioneers of approximation theory in normed linear spaces, and of generalizations of approximation theory to optimization theory. He has been a Visiting Professor at several universities in the U.S.A., Great Britain, Germany, Holland, Italy, and other countries, and was the principal speaker at an N. S. F. Regional Conference at Kent State University. He is one of the editors of the journals Numerical Functional Analysis and Optimization (since its inception in 1979), Optimization, and Revue d'analyse num'erique et de th'eorie de l'approximation. His previous books include Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer 1970), The Theory of Best Approximation and Functional Analysis (SIAM 1974), Bases in Banach Spaces I, II (Springer, 1970, 1981), and Abstract Convex Analysis (Wiley-Interscience, 1997).

Convex functions. --- Convex sets. --- Duality theory (Mathematics) --- Approximation theory. --- Convex domains. --- Convexity spaces. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Spaces, Convexity --- Convex sets --- Vector spaces --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Algebra --- Mathematical analysis --- Topology --- Sets, Convex --- Convex domains --- Set theory --- Functions, Convex --- Functions of real variables --- Operator theory. --- Functional analysis. --- Mathematical optimization. --- Mathematics. --- Operator Theory. --- Functional Analysis. --- Optimization. --- Approximations and Expansions. --- Math --- Science --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Functional calculus --- Functional equations --- Integral equations

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Kunst en natuur mathematische modellen fractalen --- Kunst en geometrie --- 7.013 --- Kunst verhouding, vorm, ritme --- Fractales --- Modèle mathématique --- 517.987 --- 514.17 --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Measures. Representations of Boolean algebras. Metric theory of dynamic systems --- Convex sets. Geometric figure arrangements. Geometric inequalities --- Fractals. --- Mathematics (General) --- 514.17 Convex sets. Geometric figure arrangements. Geometric inequalities --- 517.987 Measures. Representations of Boolean algebras. Metric theory of dynamic systems --- Mathematics (General). --- Fractals --- Geometry --- Mathematical models --- Stochastic processes --- #ABIB:altk --- #TELE:SISTA --- Random processes --- Probabilities --- Models, Mathematical --- Simulation methods --- Mathematics --- Euclid's Elements --- Kunst en natuur ; mathematische modellen ; fractalen --- Kunst ; verhouding, vorm, ritme --- #WSCH:AAS2 --- #WSCH:MODS --- Geometry. --- Mathematical models. --- Stochastic processes. --- Basic Sciences. Mathematics --- Statistical methods --- Probability. --- Mathematics. --- Géométrie --- Modèles mathématiques --- Processus stochastiques --- Geometric measure theory --- Mesure géométrique, Théorie de la --- Fractales. --- Geometric probabilities. --- Probabilités géométriques. --- Géométrie analytique --- Mathématiques --- Fractal geometry of nature --- Probabilite --- Turbulence --- Fractale geometry --- Geometrie --- Geometrie fractale --- Theorie geometrique de la mesure --- Mesure géométrique, Théorie de la --- Probabilités géométriques.