Cauchy’s surface area formula says that for a convex body K in n-dimensional Euclidean space, the mean value of the \((n-1)\)-dimensional volumes of the orthogonal projections of K to hyperplanes is a constant multiple of the surface area of K. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation on convex bodies.

中文翻译：

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柯西表面积公式的矢量类似物

柯西表面积公式表示，对于n维欧几里得空间中的凸体K ， K到超平面的正交投影的\((n-1)\)维体积的平均值是表面积的常数倍K的面积。我们证明了一个类似的公式，其中投影的体积被它们的矩向量代替。这需要在凸体上引入新的向量值评估。