Abstrakt
The kth projection function v(k)(K,.) of a convex body K subset of R-d, d >= 3, is a function on the Grassmannian G(d,k) which measures the k-dimensional volume of the projection of K onto members of G(d,k). For k=1 and k=d-1, simple formulas for the projection functions exist.
In particular, v(d-1)(K,.) can be written as a spherical integral with respect to the surface area measure of K. Here, we generalize this result and prove two integral representations for v(k)(K,.), k=1,...,d-1, over flag manifolds.
Whereas the first representation generalizes a result of Ambartzumian (1987), but uses a flag measure which is not continuous in K, the second representation is related to a recent flag formula for mixed volumes by Hug, Rataj and Weil (2013) and depends continuously on K.