By Jiří Matoušek (auth.), Jiří Matoušek (eds.)

Discrete geometry investigates combinatorial houses of configurations of geometric gadgets. To a operating mathematician or desktop scientist, it bargains subtle effects and methods of serious range and it's a beginning for fields reminiscent of computational geometry or combinatorial optimization.

This e-book is essentially a textbook creation to varied components of discrete geometry. In every one zone, it explains a number of key effects and techniques, in an obtainable and urban demeanour. It additionally comprises extra complex fabric in separate sections and therefore it may function a set of surveys in numerous narrower subfields. the most issues comprise: fundamentals on convex units, convex polytopes, and hyperplane preparations; combinatorial complexity of geometric configurations; intersection styles and transversals of convex units; geometric Ramsey-type effects; polyhedral combinatorics and high-dimensional convexity; and finally, embeddings of finite metric areas into normed spaces.

Jiri Matousek is Professor of laptop technology at Charles college in Prague. His learn has contributed to numerous of the thought of components and to their algorithmic functions. this can be his 3rd book.

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**Example text**

For example, the well-known drawing of the Fano plane of order 3 has to contain a curved line: Recently Pinchasi [Pin02] proved the following conjecture of Bez dek, resembling Sylvester's problem: For every finite family of at least 5 unit circles in the plane, every two of them intersecting, there exists an intersection point common to exactly 2 of the circles. The problems of estimating the maximum number of point-line incidences, the maximun1 nurnber of unit distances, and the minimum number of distinct distances were raised by Erdos [Erd46] .

3 Proposition (Approximating an irrational number by a frac tion) . Let a E ( 0, 1 ) be a real number and N a natural number. Then there exists a pair of natural numbers m, n such that n < N and 1 . nN This proposition implies that there arc infinitely many pairs m, n such that Ia - : I < 1 /n2 ( Exercise 4 ) . This is a basic and well-known result in elementary number theory. It can also be proved using the pigeonhole principle. The proposition has an analogue concerning the approximation of several numbers a1 , .

For all m, n > 1 , we have l(m, n) = O(m 213 n 213 + m + n) , and this bound is asymptotically tight. 42 Chapter 4: Incidence Problems We give two proofs in the sequel, one simpler and one including techniques useful in more general situations. We will mostly consider only the most interesting case m = n. The general case needs no new ideas but only a little more complicated calculation. Of course, the problem of point-line incidences can be generalized in many ways. We can consider incidences between points and hyperplanes in higher dimensions, or between points in the plane and some family of curves, and so on.