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In this short book, the authors discuss three types of problems from combinatorial geometry: Borsuk's partition problem, covering convex bodies by smaller homothetic bodies, and the illumination problem. They show how closely related these problems are to each other. The presentation is elementary, with no more than high-school mathematics and an interest in geometry required to follow the arguments. Most of the discussion is restricted to two- and three-dimensional Euclidean space, though sometimes more general results and problems are given. Thus even the mathematically unsophisticated reader can grasp some of the results of a branch of twentieth-century mathematics that has applications in such disciplines as mathematical programming, operations research and theoretical computer science. At the end of the book the authors have collected together a set of unsolved and partially solved problems that a sixth-form student should be able to understand and even attempt to solve.

Geometry, Solid. --- Convex domains.

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This textbook presents a collection of interesting and sometimes original exercises for motivated students in mathematics. Written in the same spirit as Volume 1, this second volume of Mathematical Tapas includes carefully selected problems at the intersection between undergraduate and graduate level. Hints, answers and (sometimes) comments are presented alongside the 222 “tapas” as well as 8 conjectures or open problems. Topics covered include metric, normed, Banach, inner-product and Hilbert spaces; differential calculus; integration; matrices; convexity; and optimization or variational problems. Suitable for advanced undergraduate and graduate students in mathematics, this book aims to sharpen the reader’s mathematical problem solving abilities.

Mathematics. --- Mathematics, general. --- Math --- Science --- Convex domains. --- Combinatorial optimization. --- Calculus.

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Cossos convexos --- Dominis convexos --- Geometria diferencial --- Convex bodies. --- Convex domains

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Measure theory. --- Gaussian processes. --- Hilbert space. --- Convex domains. --- Mesure, Théorie de la --- Probabilités cylindriques. --- Measure theory --- Cylindrical probabilities. --- Mesures de probabilités --- Probability measures

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The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition. The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.

Mathematics. --- Math --- Science --- Convex surfaces. --- Polyhedra. --- Sphere. --- Geometry, Solid --- Shapes --- Orbs --- Polyhedral figures --- Polyhedrons --- Convex areas --- Convex domains --- Surfaces --- Superfícies convexes --- Poliedres --- Esfera