Flag area measures

TL;DR

This paper introduces a new class of smooth, $ ext{SO}(n)$-covariant flag area measures on Euclidean spaces, providing explicit formulas and a classification theorem akin to Hadwiger's, expanding the understanding of valuation measures.

Contribution

It constructs a general sequence of smooth $ ext{SO}(n)$-covariant flag area measures and classifies all such measures, extending previous work with explicit formulas involving principal angles.

Findings

Explicit formulas for flag area measures on polytopes.

Construction of a general sequence of smooth measures.

Classification of all smooth $ ext{SO}(n)$-covariant flag area measures.

Abstract

A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0 \leq p \leq n-1$. Using local parallel sets, Hinderer constructed examples of $\mathrm{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\mathrm{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\mathrm{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger's theorem.