Abstract Crofton's intersection formula states that the ( n − j ) th intrinsic volume of a compact convex set in R n can be obtained as an invariant integral of the ( k − j ) th intrinsic volume of sections with k-planes. This paper discusses the question if the ( k − j ) th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we show that the sums of the ( k − j ) th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k = 1 or when k = 2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k ≥ 2 .

中文翻译：

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克罗夫顿公式中测量函数的唯一性

摘要 Crofton 交集公式表明，R n 中的紧凸集的第 ( n − j ) 个固有体积可以作为具有 k 平面的截面的第 ( k − j ) 个固有体积的不变积分获得。本文讨论第(k-j)个固有体积是否可以用其他泛函代替，即Crofton公式中的测量函数是否唯一的问题。答案是否定的：我们证明第 ( k − j ) 个内在体积和某些平移不变连续估值的同质度 k 的总和产生反例。如果测量函数是局部的，那么当 k = 1 或 k = 2 时，这些是唯一的例子，我们将考虑限制为偶数测量函数。当 k ≥ 2 时，可以构造局部泛函的其他示例。