By Janos Kollar

ISBN-10: 0821804324

ISBN-13: 9780821804322

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Extra resources for Complex Algebraic Geometry (Ias Park City Mathematics Series, V. 3) PCMS 3

Example text

The rest of this book specific triangulations of them. One can, for example, completely characterize which products of two simplices possess non-regular triangulations. Finally, cubes and 0/1 polytopes, although a bit less structured in general, are very important in polyhedral optimization, so it is worth having a look at their triangulations. Chapter 7 is devoted to “pathological” triangulations. As a warm-up for later constructions, it starts with a very close look at the smallest configuration having non-regular triangulations, which we call “the mother of all examples”.

A rather easy proof can be derived by induction on the dimension of the simplex (see exercises). 32 for an example and [99, 233] for details concerning the following theorem and a strengthening). 5 (Polytopal Sperner Lemma). Let P be a d-dimensional polytope with n vertices v1 , v2 , . . , vn . Let T be a triangulation of P, possibly using additional vertices. Label the vertices of T by 1, 2, . . , n in such a way that a vertex of T belonging to a face F of P can only be labeled by j if v j is a vertex of F.

Monotone flipping does not always work, which leaves open (in dimensions three and four) the question of whether graphs of flips are always connected. • Triangulations exist with fewer flips than expected. Chapter 4 continues with the work of Chapter 2, providing the reader with a tool box of constructions and results to use when needed. 1 isolates the properties of a given configuration that are relevant to study its set of triangulations. This is basically the oriented matroid of the configuration (sometimes called its order type) which can be described in several ways.

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Complex Algebraic Geometry (Ias Park City Mathematics Series, V. 3) PCMS 3 by Janos Kollar

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