Titlebook: Lectures on Discrete Geometry; Ji?í Matou?ek Textbook 2002 Springer-Verlag New York 2002 Discrete Geometry.Dimension.discrete geometry.geo

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Lower Envelopes, Already for segments in the plane, this problem has an unexpectedly subtle and difficult answer. The closely related combinatorial notion of . has proved to be a useful general tool, since the surprising phenomena encountered in the analysis of the lower envelope of segments are by no means rare in

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Intersection Patterns of Convex Sets,one closely related but more difficult theorem in the current chapter. These more advanced relatives are selected, among the vast number of variations on the Helly-Radon-Carathéodory theme, because of their wide applicability and also because of nice techniques and tricks appearing in their proofs.

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DUST

Two Applications of High-Dimensional Polytopes,ing geometric objects in arbitrary dimension, we could mostly rely on the intuition from the familiar dimensions 2 and 3. In the present chapter we can still use dimensions 2 and 3 to picture examples, but these tend to be rather trivial. For instance, in the first section we are going to prove thin

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Volumes in High Dimension,miliar” bodies behave quite differently in high dimensions from what the 3-dimensional intuition suggests. Then we calculate that any convex polytope in the unit ball .. whose number of vertices is at most polynomial in . occupies only a tiny fraction of .. in terms of volume. This has interesting c

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Measure Concentration and Almost Spherical Sections,s that if . ? ... is a set occupying at least half of the sphere, then almost all points of ... are quite close to ., at distance about .(..). Measure concentration is an extremely useful technical tool in high-dimensional geometry. From the point of view of probability theory, it provides tail esti

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Ji?í Matou?ekeggs (2, 3). Nevertheless, the chronology of these events does not mean that nonhuman primate IVF cannot point the way to significant improvements in human IVF technology nor increase understanding of key areas in human reproduction. Rather, the indication is that IVF technology in nonhuman primates

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