Investigation of Convex Bodies in R3 by Support Planes

Abstract

Abstract: Let R3 be the 3-dimensional Euclidean space and D be a bounded convex body (Formula presented.). Consider a family of support planes for which D is an envelope. How can we obtain information about D from the support planes? Conditions under which a given convex body is the envelope of a family of planes are obtained. Therefore the distances of these planes from the origin will be the support functions of this body D. In particular, we have cited expressions for the surface area and the volume of body D in terms of support planes.