By Herbert Busemann

Starting with a quick evaluate of notations and terminology, the textual content proceeds to convex curves, the theorems of Meusnier and Euler, extrinsic Gauss curvature, and the impact of the curvature at the neighborhood form of a floor. A bankruptcy at the Brunn-Minkowski conception and its purposes is through examinations of intrinsic metrics, the metrics of convex hypersurfaces, geodesics, angles, triangulations, and the Gauss-Bonnet theorem. the ultimate bankruptcy explores the tension of convex polyhedra, the conclusion of polyhedral metrics, Weyl's challenge, neighborhood attention of metrics with non-negative curvature, open and closed surfaces, and smoothness of realizations.

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**Example text**

These relationships among pencils of planes, lines, and points provide the basis for perspective and projective correspondences (Sections 2-5 and 2-6). Throughout this section the words "point" and "plane" have been interchanged in statements regarding points and planes to obtain new statements. Also the words" point" and "line" have been interchanged in statements regarding points and lines on a projective plane. The resulting pairs of statements illustrate the principles of duality (Section 2-4).

C: c~ '7 FIG. 2-6 FIG. 2- 7 Given any space figure F and any point P that is not a point of F, the points and lines of F determine with P a set of lines and planes called the point section of the space figure F by P. For example, if the figure F is taken as triangle ABC and if P is not on the plane ABC (Fig. 2-6), then the point section of F by P consists of the lines PA, PB, PC and the planes PAB, PAC, PBC. Given any space figure F and any line m that is not a line of F, the points, lines, and planes of F determine with m a set of planes and points called the line section of the space figure F by m.

If either of the lines coincides with the line AB, they intersect, as proved in the proof of Theorem 2-1. Thus we may assume that F ~ S and that the lines FG and RS are both distinct from AB. Then, since the line FG has at most one point in common with the line AB, is determined by any two of its points (Exercise 4, Section 2-1) , and contains at least three points, we may assume that the points F and G are not on AB. 2-21 PROPERTIES OF A PROJECTIVE PLANE 3] AB J FIG. 2-4 FIG. 2-5 Similarly, we may assume that the points Rand S are not on AB.