Convex Body

a geometric body having the property that any segment joining two of its points is entirely contained within it. The body shown in Figure l,a is convex and that shown in Figure l,b is not. A sphere, a cube, a spherical segment, or a half-space are examples of convex bodies. Any connected part of a boundary of a convex body is called a convex surface. Through every point of the boundary of a convex body passes at least one plane of support having a point (or segment or part of a plane) in common with the boundary of the body but not intersecting it (plane P in Figure l,a). At points where the boundary of the convex body is a smooth surface, the plane of support will be tangential. At points where the smoothness of the surface is violated (as at the corner of a cube), an infinite number of planes of support can be drawn.

Figure 1

There are five types of convex bodies: finite (the boundary is a closed convex surface), infinite (the boundary is one infinite surface; for example, a convex body bounded by a paraboloid), cylinders infinite in both directions (the boundary is a closed convex cylindrical surface; for example, an infinite circular cylinder), layers between pairs of parallel surfaces; and all space. Convex bodies can be defined by a support function expressing the distance from the origin of coordinates to the support surface as a function of the outer normal to the convex body (that is, of a unit vector that is perpendicular to the support surface and directed toward that one of two half-spaces that is defined by this surface and which contains no points of the convex body).

Convex polyhedrons are the simplest convex bodies; they are convex bodies bounded by a finite number of polygons. For any finite convex body it is possible to construct polyhedrons approximating it to any degree. This makes it possible to solve many problems of convex bodies as follows: the problem is solved for convex polyhedrons, and then by going to the limit, the corresponding result is established for any convex body. For example, the areas of convex surfaces and the volumes of any convex bodies are determined in this way. In particular, it can be proved that if one finite convex body encompasses another, then the surface area of the first is greater than the surface area of the second. The method described was worked out in detail by A. D. Aleksandrov and is used to solve a variety of new problems in the theory of convex bodies.

The general theory of convex bodies and convex surfaces constitutes the so-called geometry of convex bodies. Problems in the geometry of convex bodies cover a wide range of questions: the general properties of convex bodies (theorems on support surfaces, classification of convex bodies, and approximation by polyhedrons), extreme properties of convex bodies (for example, of all convex bodies with a given volume, the sphere has the minimum surface), existence and uniqueness theorems about convex bodies with given proper-ties (for example, the theorem on the existence of a convex polyhedron with given directions and areas of faces), properties of various classes of convex bodies (for example, of bodies of constant width), general properties of convex sur-faces, existence and uniqueness theorems for convex surfaces, the internal geometry of convex surfaces, and others. The concept of convex bodies naturally arises in the geometry of spaces of constant curvature. Many of the problems listed above have been formulated and solved for convex bodies in such spaces. The methods and results of the theory of convex bodies are used in various branches of mathematics: in geometry, the theory of numbers, and mathematical analysis. The principles of the theory of convex bodies were established in the late 19th century by the German mathematicians H. Brunn and H. Minkowski. Major new results of this theory have been obtained by Soviet mathematicians A. D. Aleksandrov and A. V. Pogorelov.