Equality of two representations of extended affine surface area

Abstract

1. Introduction and statement of the Theorem. In the last decade there has been growing interest in concepts and theorems related to the equiaffine surface area of a convex body. Whereas originally the notion of an equiaffinety invariant surface area was limited to the scope of affine differential geometry developed by Blaschke and his school, recent research is mainly devoted to determining and investigating corresponding expressions which are applicable to the boundary of an arbitrary convex body without imposing further smoothness or curvature assumptions. Interest in affine surface area is partly due to its connections with various geometric inequalities such as the Blaschke-Santal6 inequality [9]. See Lutwak [7] for a survey. Some motivation can also be derived from its significance in the context of polyhedral approximation [10], [13], [2] or in the investigation of floating bodies [14], [15]. Recently, three substantially different definitions for the equiaffine surface area of an arbitrary convex body have been proposed by Lutwak [6], LeichtweiB [4], and Schfitt and Werner [14]. It is a natural task to investigate how these definitions are related to each other. Before we describe the main results, some definitions are required. In the following we work in Euclidean d-space, 11 a, with scalar product (. ,. ) and norm l" 1- Let d > 2 and denote by o~( ~ the class of all nonempty compact, convex sets of IRa. As usual, set S a- 1 : = {x ~ lRa l t x I = 1 ). We write [o, x] for the closed line segment joining o and x, A star body in R n is a nonempty compact set L c Ra satisfying [o, x] c L for all x ~ L and such that the radial function QL: S d- 1 ~ R, defined by