A flag representation of projection functions

TL;DR

This paper introduces new integral representations for projection functions of convex bodies over flag manifolds, extending previous formulas and providing continuous dependencies on the convex body.

Contribution

The paper generalizes known formulas for projection functions to all dimensions using flag measures, including a continuous representation related to recent mixed volume formulas.

Findings

Derived two integral representations for projection functions over flag manifolds.

Connected the second representation to recent mixed volume flag formulas.

Provided a continuous dependence of the representation on the convex body.

Abstract

The $k$th projection function $v_k(K,\cdot)$ of a convex body $K\subset {\mathbb R}^d, d\ge 3,$ 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, $v_{d-1}(K,\cdot)$ 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 $v_k(K,\cdot), k=1,\dots,d-1$, 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 $K$.