A Characterization of Affine Surface Area

A Characterization of Affine Surface Area

We show that every upper semicontinuous and equi-affine invariant valuation on the space of d-dimensional convex bodies is a linear combination of affine surface area, volume and the Euler characteristic. 1991 AMS subject classification: Primary 52A20; Secondary 53A15.

Inequalities for mixed p - affine surface area ∗

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of Lp affine surface areas, mixed p-affine surface areas and other ...

Random Polytopes and Affine Surface Area

Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. E(K,n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with C3 boundary and everywhere positive curvature c(d) lim n→∞ vold(K)− E(K,n) ( vold(K) n ) 2 d+1 = ∫ ∂K κ(x) 1 d+1 dμ(x) where κ(...

On the p-affine surface area

A general geometric construction for affine surface area

Let K be a convex body in R and B be the Euclidean unit ball in R. We show that limt→0 |K| − |Kt| |B| − |Bt| = as(K) as(B) , where as(K) respectively as(B) is the affine surface area of K respectively B and {Kt}t≥0, {Bt}t≥0 are general families of convex bodies constructed from K, B satifying certain conditions. As a corollary we get results obtained in [M-W], [Schm],[S-W] and[W]. The affine su...