General Affine Surface Areas

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established. 2000 AMS subject classification: Primary 52A20; Secondary 53A15. Finding the right notion of affine surface area was one of the first questions asked within affine differential geometry. At the beginning of the last century, Blaschke [5] and his School studied this question and introduced equi-affine surface area – a notion of surface area that is equi-affine invariant, that is, SL(n) and translation invariant. The first fundamental result regarding equi-affine surface area was the classical affine isoperimetric inequality of differential geometry [5]. Numerous important results regarding equi-affine surface area were obtained in recent years (see, for example, [1,2,45,48–51]). Using valuations on convex bodies, the author and Reitzner [27] were able to characterize a much richer family of affine surface areas (see Theorem 2). Classical equi-affine and centro-affine surface area as well as all Lp affine surface areas for p > 0 belong to this family of Lφ affine surface areas. The present paper has two aims. The first is to establish affine isoperimetric inequalities and basic duality relations for all Lφ affine surface areas. The second aim is to define new general notions of affine surface area that complement Lφ affine surface areas and include Lp affine surface areas for p < −n and −n < p < 0. Let Kn 0 denote the space of convex bodies, that is, compact convex sets, in Rn that contain the origin in their interiors. Whereas Lφ affine surface areas are always finite and are upper semicontinuous functionals on Kn 0 , the affine surface areas of the new families are infinite for certain convex bodies including polytopes and are lower semicontinuous functionals on Kn 0 . Basic properties and affine isoperimetric inequalities for these new affine surface areas are established. In Section 6, ∗Research supported, in part, by NSF grant DMS-0805623