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**Geometry. I, II. Transl. from the French by M. Cole and S. Levy.**
*(English)*
Zbl 0606.51001

Universitext. Berlin etc.: Springer-Verlag. I: XIII, 428 p.; DM 74.00; II: X, 406 p.; DM 74.00 (1987).

This two-volume textbook is the English translation of the five-volume French original [Vol. 1. (1977; Zbl 0382.51011); Vol. 2. (1977; Zbl 0382.51012); Vol. 3. (1978; Zbl 0423.51001); Vol. 4. (1978; Zbl 0423.51002); Vol. 5. (1977; Zbl 0423.51003)].

The contents include the main fields of elementary geometry, and of convex geometry, e.g. affine and projective geometry, Euclidean spaces, triangles, spheres, circles, convex sets, polytopes, projective and affine quadrics, projective and Euclidean conics, elliptic and hyperbolic geometry, and spherical geometry.

In addition, the author deals with special topics in convex geometry and Euclidean space, e.g. the classification of regular polytopes, the isoperimetric inequality, the theorem of Helly and Krasnosel’skij, Poncelet’s theorem, and Cauchy’s theorem. These results are often absent from comperable books.

As the author points out, his main aim is the motivation and illustration of given definitions, notations and results. This kind of illustration is highly recommendable and specific for this textbook. The book is a rich source for exercises (most of them more difficult than those in comparable books. See the companion volume of problems and solutions by the author, P. Pansu, J.-P. Berry, and X. Saint- Raymond, ”Problems in Geometry” (1984; Zbl 0543.51001), French original in (1982; Zbl 0489.51001)), for problems and its extensive bibliography.

The contents include the main fields of elementary geometry, and of convex geometry, e.g. affine and projective geometry, Euclidean spaces, triangles, spheres, circles, convex sets, polytopes, projective and affine quadrics, projective and Euclidean conics, elliptic and hyperbolic geometry, and spherical geometry.

In addition, the author deals with special topics in convex geometry and Euclidean space, e.g. the classification of regular polytopes, the isoperimetric inequality, the theorem of Helly and Krasnosel’skij, Poncelet’s theorem, and Cauchy’s theorem. These results are often absent from comperable books.

As the author points out, his main aim is the motivation and illustration of given definitions, notations and results. This kind of illustration is highly recommendable and specific for this textbook. The book is a rich source for exercises (most of them more difficult than those in comparable books. See the companion volume of problems and solutions by the author, P. Pansu, J.-P. Berry, and X. Saint- Raymond, ”Problems in Geometry” (1984; Zbl 0543.51001), French original in (1982; Zbl 0489.51001)), for problems and its extensive bibliography.

Reviewer: G.Ehrig

### MSC:

51-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry |

52-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry |

51M05 | Euclidean geometries (general) and generalizations |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

51N10 | Affine analytic geometry |

51N15 | Projective analytic geometry |

51N20 | Euclidean analytic geometry |

52Bxx | Polytopes and polyhedra |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

52A35 | Helly-type theorems and geometric transversal theory |