Problems in Discrete and Combinatorial Geometry

Publisher Summary This chapter presents an overview of the problems observed in discrete and combinatorial geometry. The term combinatorial geometry seems to have been coined by H. Hopf and was made popular by the book of Hadwiger, Debrunner and Klee and others. Combinatorial geometry closely resembles another field named elementary number theory. In fact, methods of analytic number theory are often used to treat a geometric problem. Almost every question considered in convexity has its discrete or combinatorial aspects. This fact explains the great diversity of questions that can be classified as combinatorial or discrete, and the equally large diversity of methods used to deal with them.

[1]  On Partitions of an Equilateral Triangle , 1967, Canadian Journal of Mathematics.

[2]  Karol Borsuk.  Drei Sätze über die n-dimensionale euklidische Sphäre , 1933 .

[3]  Noga Alon,et al.  The maximum size of a convex polygon in a restricted set of points in the plane , 1989, Discret. Comput. Geom..

[4]  W. T. Tutte,et al.  Squaring the Square , 1950, Canadian Journal of Mathematics.

[5]  J. Seidel.  Graphs and two-distance sets , 1981 .

[6]  P. Federico.  Some simple perfect 2×1 rectangles , 1970 .

[7]  F. Levi.  Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns , 1955 .

[8]  Boris V. Dekster.  Diameters of the pieces in Borsuk's covering , 1989 .

[9]  A. Seidenberg,et al.  A Simple Proof of a Theorem of Erdös and Szekeres , 1959 .

[10]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[11]  W. Bonnice,et al.  On Convex Polygons Determined by a Finite Planar Set , 1974 .

[12]  Peter Ungar,et al.  2N Noncollinear Points Determine at Least 2N Directions , 1982, J. Comb. Theory, Ser. A.

[13]  Marek Lassak.  Solution of Hadwiger's Covering Problem for Centrally Symmetric Convex Bodies in E3 , 1984 .

[14]  Über die Zerstückung eines Eikörpers , 1948 .

[16]  N. Kazarinoff,et al.  On existence of compound perfect squared squares of small order , 1973 .

[17]  Heiko Harborth.  Konvexe Fünfecke in ebenen Punktmengen. , 1978 .

[18]  James B. Shearer,et al.  Tiling rectangles with rectangles , 1982 .

[19]  Aart Blokhuis.  A New Upper Bound for The Cardinality of 2-Distance Sets in Euclidean Space , 1981 .

[20]  P. J. Federico.  THE NUMBER OF POLYHEDRA , 1975 .

[21]  On the problem to cover simply and without gap the inside of a square with a finite number of squares which are all different from one another , 1932 .

[22]  Nicholas D. Kazarinoff,et al.  Squaring Rectangles and Squares , 1973 .

[23]  S. K. Stein,et al.  Equidissections of polygons , 1990, Discret. Math..

[24]  P. Révész,et al.  Zum Borsukschen Zerteilungsproblem , 1956 .

[25]  O. Schramm.  Illuminating Sets of Constant Width , 1988 .

[26]  Joseph Malkevitch.  TILING CONVEX POLYGONS WITH EQUILATERAL TRIANGLES AND SQUARES , 1985 .

[27]  Ludwig Danzer,et al.  Three-dimensional analogs of the planar penrose tilings and quasicrystals , 1989, Discret. Math..

[28]  Sherman K. Stein.  Equidissections of centrally symmetric octagons , 1989 .

[29]  B. Grünbaum.  A simple proof of Borsuk's conjecture in three dimensions , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  P. Duchet,et al.  Sous Les Pavés , 1983 .

[31]  R. J. M. Dawson.  On Filling Space with Different Integer Cubes , 1984, J. Comb. Theory, Ser. A.

[32]  J. Kahn,et al.  A counterexample to Borsuk's conjecture , 1993, math/9307229.

[33]  C. A. Rogers.  Symmetrical sets of constant width and their partitions , 1971 .

[34]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[35]  J. Bólyai,et al.  SOME APPLICATIONS OF GRAPH THEORY AND COMBINATORIAL METHODS TO NUMBER THEORY AND GEOMETRY , 1978 .

[36]  Borsuk's covering for blunt bodies , 1988 .

[37]  P. Scott.  On the Sets of Directions Determined by n Points , 1970 .

[38]  E. A. Kasimatis.  Dissections of regular polygons into triangles of equal areas , 1989, Discret. Comput. Geom..

[39]  Tibor Bisztriczky,et al.  Convexly independent sets , 1990, Comb..

[40]  Ilona Palásti.  A distance problem of P. Erdös with some further restrictions , 1989, Discret. Math..

[41]  W. T. Tutte.  The Quest of the Perfect Square , 1965 .

[42]  Thomas W. Cusick,et al.  View-Obstruction Problems in n-Dimensional Geometry , 1974, J. Comb. Theory, Ser. A.

[43]  Richard Pollack,et al.  On the Combinatorial Classification of Nondegenerate Configurations in the Plane , 1980, J. Comb. Theory, Ser. A.

[44]  H. Eggleston.  Covering a Three‐Dimensional set with Sets of Smaller Diameter , 1955 .

[45]  John Thomas.  A Dissection Problem , 1968 .

[46]  Zalman Usiskin,et al.  Can Every Triangle Be Divided into n Triangles Similar to It , 1970 .

[47]  H. Hadwiger.  Überdeckung einer Menge durch Mengen kleineren Durchmessers , 1945 .

[48]  Ronald L. Graham.  Fault-free Tilings of Rectangles , 1981 .

[49]  Joseph Malkevitch.  3‐VALENT 3‐POLYTOPES WITH FACES HAVING FEWER THAN 7 EDGES , 1970 .

[50]  John K. Williams,et al.  Rep-tiling for triangles , 1991, Discret. Math..

[51]  P. Erdös.  PROBLEMS AND RESULTS IN COMBINATORIAL GEOMETRY a , 1985 .

[52]  Noga Alon,et al.  Covering a square by small perimeter rectangles , 1986, Discret. Comput. Geom..

[53]  Paul Monsky.  On Dividing A Square Into Triangles , 1970 .

[54]  Arie Bialostocki,et al.  Some notes on the Erdös-Szekeres theorem , 1990, Discret. Math..

[55]  J. J. Seidel,et al.  On Two-Distance Sets in Euclidean Space , 1977 .

[56]  Covering a plane convex body by four homothetical copies with the smallest positive ratio , 1986 .

[57]  Raul Cordovil,et al.  The directions determined by n points in the plane: a matroidal generalization , 1983, Discret. Math..

[59]  Paul Erdös.  Some new problems and results in Graph Theory and other branches of Combinatorial Mathematics , 1981 .

[60]  I. Bárány,et al.  Empty Simplices in Euclidean Space , 1987, Canadian Mathematical Bulletin.

[61]  George B. Purdy,et al.  The Directions Determined by n Points in the Plane , 1979 .

[62]  Peter J. Robinson.  Fault-free rectangles tiled with rectangular polyominoes , 1982 .

[63]  David Gale,et al.  On inscribing $n$-dimensional sets in a regular $n$-simplex , 1953 .

[64]  A. Meir,et al.  On empty triangles determined by points in the plane , 1988 .

[65]  SOLUTION OF HADWIGER'S PROBLEM FOR A CLASS OF CONVEX BODIES , 1991 .

[66]  Robert E. Jamison,et al.  Planar configurations which determine few slopes , 1984 .

[67]  W. T. Tutte.  The dissection of equilateral triangles into equilateral triangles , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[68]  H. Lenz.  Zur Zerlegung von Punktmengen in solche kleineren Durchmessers , 1955 .

[69]  M. Laczkovich.  Tilings of polygons with similar triangles , 1990, Comb..

[70]  Robert E. Jamison.  Direction trees , 1987, Discret. Comput. Geom..

[71]  Paul Erdös,et al.  A problem of Leo Moser about repeated distances on the sphere , 1989 .

[72]  A. J. W. Duijvestijn,et al.  Simple perfect squared square of lowest order , 1978, J. Comb. Theory, Ser. B.

[73]  K. Bezdek,et al.  Hadwiger-Levi’s Covering Problem Revisited , 1993 .

[74]  Leo Moser,et al.  On The Different Distances Determined By n Points , 1952 .

[75]  Jasper Dale Skinner.  Uniquely squared squares of a common reduced side and order , 1992, J. Comb. Theory, Ser. B.

[76]  J. Horton.  Sets with No Empty Convex 7-Gons , 1983, Canadian Mathematical Bulletin.

[77]  Scott Johnson.  A new proof of the Erdos-Szekeres convex k-gon result , 1986, J. Comb. Theory, Ser. A.

[78]  Tibor Bisztriczky,et al.  Nine convex sets determine a pentagon with convex sets as vertices , 1989 .

[79]  R. Sprague.  Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate , 1939 .

[80]  M. Dehn.  Über Zerlegung von Rechtecken in Rechtecke , 1903 .

[81]  Marek Lassak.  Covering the boundary of a convex set by tiles , 1988 .

[82]  Robert E. Jamison,et al.  A SURVEY OF THE SLOPE PROBLEM , 1985 .

[83]  C. Bouwkamp,et al.  Tables relating to simple squared rectangles of orders nine through fifteen , 1961 .

[84]  Paul Erdös,et al.  Some Old and New Problems in Combinatorial Geometry , 1984 .

[85]  H. T. Croft.  9-Point and 7-Point Configurations in 3-Space , 1962 .

[86]  H. Hadwiger.  Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers , 1946 .

[87]  A. Clivio.  Tilings of a torus with rectangular boxes , 1991, Discret. Math..

[88]  W. T. Tutte,et al.  The Dissection of Rectangles Into Squares , 1940 .

[89]  J. Seidel,et al.  Spherical codes and designs , 1977 .