The illumination body of almost polygonal bodies

The affine surface area was introduced by Blaschke [B] for convex bodies in R with sufficently smooth boundary and by Leichtweiss [L1] for convex bodies in R with sufficently smooth boundary. As it occurs naturally in many important questions so for example in the approximation of convex bodies by polytopes ( see the survey article of Gruber [G] ) or in a priori estimates for PDEs [Lu2] one wanted to have extensions of the affine surface area to arbitrary convex bodies in R without any smoothness assumptions of the boundary. Such extensions were given in recent years by Leichtweiss [L2], Lutwak[Lu1], Schuett and Werner [S-W1] and Werner [W1]. In many of these extensions the affine surface area is defined using the volume differences voln(K) voln(Kδ) respectively voln(K ) voln(K) where Kδ is a floating body ( see [L1] and [S-W1] for the definition ) and Kis an illumination body. Therefore one is interested in the precise behaviour of such volume differences. We give here the analogue for illumination bodies to results obtained in [SW2] for floating bodies; namely we investigate what kind of functions can occur as the difference volume vol2(K ) vol2(K) for convex bodies in R. The result and the methods of proof are similar to [S-W2].