# Grassmann measures of convex bodies

## TL;DR

This paper introduces Grassmann measures derived from flag measures to uniquely identify convex bodies and explores their role in valuation theory, projection averages, and touching measures, advancing geometric analysis of convex bodies.

## Contribution

It defines Grassmann measures as images of flag measures, proving their uniqueness for convex bodies and linking them to valuation representations and geometric measures.

## Key findings

- Grassmann measures uniquely determine centrally symmetric convex bodies.
- They are connected to smooth, translation-invariant, even valuations.
- The paper establishes a uniqueness result for projection averages of area measures.

## Abstract

Flag measures are descriptors of convex bodies $K$ in $d$-dimensional Euclidean space generalizing the classical area measures. They have been used to provide general integral formulas for mixed volumes (see Hug, Rataj and Weil (2017)). Here, we consider an image measure $\gamma_j(K,\cdot)$ of flag measures, defined on the Grassmannian $G(d,j)$ of affine $j$-spaces, $1\le j\le d-1$, and show that it determines centrally symmetric bodies $K$ of dimension $\geq j+1$ uniquely. We then explain that Grassmann measures appear in the representation of smooth, translation invariant, continuous and even valuations due to Alesker (2003). Using this connection, we prove a uniqueness result for projection averages of area measures and we finally discuss a Grassmann version of the natural touching measure of convex bodies.