Una Fórmula de Steiner para superfícies paralelas en geometría afín

Abstract

El objeto principal de esta nota es generalizar a la geometría afín unimodular la fórmula de Steiner que, en el caso métrico da el volumen limitado por una superficie paralela a otra superficie cerrada S en función del volumen V limitado por S, el área F, la curvatura media total M y la curvatura total C de esta última superficie.

The main purpose of this note is to prove that with a convenient definition of parallel surfaces, the Steiner's formula which gives the volume bounded by a closed surface S» parallel to a given orientable and closed surface S may be generalized to the affine unimodular geometry, and takes the same form (4.5) that in the metrical case (S2, M, G being now the affine area, Integrated affine mean curvature and integrated affine total curvature respectively). From this result certain inequalities, dues to Blaschke, (n' 5), follows by direct application of the Brunn-Minkowski theorem on mixed volumes. In nº 6 we consider the family of affine invariants Ja (6.4) and certain inequalities (6.8), (6.10), (6.11) between the affine area, volume and maximal affine width of a convex body. In nº 7, 8 analogous questions for the case of the plane are considered