A flag representation of projection functions

Abstract

Abstract The kth projection function υk (K, ·) of a convex body K ⊂ 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, υ 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 υk (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.