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**Unsolved problems in geometry.**
*(English)*
Zbl 0748.52001

Problem Books in Mathematics 2. New York etc.: Springer-Verlag (ISBN 978-0-387-97506-1/hbk; 978-1-4612-6962-5/pbk; 978-1-4612-0963-8/ebook). xv, 198 p. (1991).

This book had its beginnings thirty years ago when the first author started to collect and circulate many unsolved problems in geometry. There was an initial large collection followed by several addenda. Although poorly printed (some problems in “scribbles”) they were received with gratitude, copied and further circulated. The world of discrete geometers waited (and waited) for the book version to appear. There was a well-known conjecture that it would not appear in this century. But it has! Halleluya!! This volume is a source book for anyone wishing to pursue problems in convex, discrete and combinatorial geometry. Following the Preface there is a list of Other Problem Collections (17 entries) and a list of Standard References (26 entries). Then follow 6 sections, subdivided into subsections each containing a problem or a group of related problems ending with references for the subsection. The reader (of this review) may get a “feel” for the content of the book from the next part of the review which gives the titles of the sections, followed by the number of subsections and the titles of 3 “sample” subsections.

A. Convexity (38): A5. Illumination problems; A14. Rotating polyhedra; A17. Isoperimetric inequalities and extremal problems. B. Polygons, Polyhedra and Polytopes (25): B10. Shadows of polyhedra; B14. Rigidity of frameworks; B20. Lengths of paths on polyhedra. C. Tiling and Dissection (20): C2. Squaring the square; C14. Which polygons tile the plane?; C20. Problems on equidecomposability. D. Packing and Covering (18): D3. Covering a circle with equal discs; D7. The problem of Tammes; D10. Packing balls in space. E. Combinatorial Geometry (14) E7. Neighborly convex bodies; E11. Sets that can be moved to cover several lattice points; E13. Variations on Minkowski’s theorem. F. Finite Sets of Points (17): F1. Minimum number of distinct distances: F4. Can each distance occur a different number of times?; F12. Lines through sets of points. G. General Geometric Problems (16): G4. Maximal sets avoiding certain distance configurations; G12. Euclidean Ramsey problems; G16. Unions of similar copies of sets.

In spite of the several decade gestation period there is evidence that the “final” manuscript was prepared in haste from the vast amount of information available. Readers should report errors (e.g., on page 150, \(r(4)=3\) should be \(r(4)=6\)) and new relevant results (e.g., a paper by Csima and Sawyer, to appear shortly in Discrete and Computational Geometry, is an important addition to F12) to the authors for consideration in preparing the second edition. Meanwhile, choose a problem from this wonderful collection and get cracking. And give thanks to C-F-G for this service to the community.

A. Convexity (38): A5. Illumination problems; A14. Rotating polyhedra; A17. Isoperimetric inequalities and extremal problems. B. Polygons, Polyhedra and Polytopes (25): B10. Shadows of polyhedra; B14. Rigidity of frameworks; B20. Lengths of paths on polyhedra. C. Tiling and Dissection (20): C2. Squaring the square; C14. Which polygons tile the plane?; C20. Problems on equidecomposability. D. Packing and Covering (18): D3. Covering a circle with equal discs; D7. The problem of Tammes; D10. Packing balls in space. E. Combinatorial Geometry (14) E7. Neighborly convex bodies; E11. Sets that can be moved to cover several lattice points; E13. Variations on Minkowski’s theorem. F. Finite Sets of Points (17): F1. Minimum number of distinct distances: F4. Can each distance occur a different number of times?; F12. Lines through sets of points. G. General Geometric Problems (16): G4. Maximal sets avoiding certain distance configurations; G12. Euclidean Ramsey problems; G16. Unions of similar copies of sets.

In spite of the several decade gestation period there is evidence that the “final” manuscript was prepared in haste from the vast amount of information available. Readers should report errors (e.g., on page 150, \(r(4)=3\) should be \(r(4)=6\)) and new relevant results (e.g., a paper by Csima and Sawyer, to appear shortly in Discrete and Computational Geometry, is an important addition to F12) to the authors for consideration in preparing the second edition. Meanwhile, choose a problem from this wonderful collection and get cracking. And give thanks to C-F-G for this service to the community.

Reviewer: W.Moser (Montreal)

### MSC:

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

00A07 | Problem books |

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\textit{H. T. Croft} et al., Unsolved problems in geometry. New York etc.: Springer-Verlag (1991; Zbl 0748.52001)

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### Online Encyclopedia of Integer Sequences:

Number of polyhedra (or 3-connected simple planar graphs) with n nodes.Sylvester’s problem: minimal number of ordinary lines through n points in the plane.

Mrs. Perkins’s quilt: smallest coprime dissection of n X n square.

Number of 4-dimensional polytopes with n vertices.

a(n) = minimal integer m such that an m X m square contains non-overlapping squares of sides 1, ..., n (some values are only conjectures).

Number of simplices in minimal decomposition of an n-cube.

Simplexity of the n-cube: minimal cardinality of triangulation of n-cube using n-simplices whose vertices are vertices of the n-cube.

Number of simplices in minimal corner-slicing triangulation of n-cube.

Numbers k such that there is an arrangement of k points in the plane such that no point is on >= k/2 connecting lines.

Numbers of ways in which a unit disc can be dissected into 6n curvilinear triangles, at least one of which does not contain the center

Decimal expansion of DeVicci’s tesseract constant.