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.

中文翻译：

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旗区测量

$n$ 维欧几里得向量空间上的标志面积测度是一个连续平移不变的估值，标志流形上的带符号测度空间中的值由单位向量 $v$ 和 $(p+1)$ 组成维线性子空间包含 $v$ 和 $0 \leq p \leq n-1$。使用局部并行集，Hinderer 构建了 $\mathrm{SO}(n)$-协变标志区域度量的示例。他在多胞体上评估的旗帜面积度量有一个明确的公式，其中涉及两个子空间之间夹角的平方余弦。我们通过对适当微分形式的正常循环进行积分，构建了一个更一般的平滑 $\mathrm{SO}(n)$-协变标志区域度量序列。我们明确描述了我们对多胞体的措施，它涉及两个子空间之间主角度的平方余弦中的任意基本对称多项式。此外，我们表明这些标志区域度量跨越了所有平滑的 $\mathrm{SO}(n)$-协变标志区域度量的空间，这给出了符合 Hadwiger 定理精神的分类结果。