We prove that in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric rank. The reconstruction takes the form of a (potentially fractional) linear PDE given in closed form. While this relation holds in the sense of distributions for an arbitrary probability measure, when it admits a density we provide sufficient conditions to ensure that the density can be recovered pointwise through the PDE. Surprisingly, the reconstruction procedure is of a local nature when the dimension is odd, and of a non-local nature in even dimensions. We give examples of the reconstruction in dimension 2 and 3. We use our results to characterise the regularity of depth contours. We conclude the paper with a partial counterpart to the non-localisability in even dimensions.

翻译：我们证明，在任意欧几里得空间中，任何概率测度均可通过其几何秩显式重建。该重建以闭合形式的（可能为分数阶）线性偏微分方程表示。虽然此关系在分布意义上对任意概率测度成立，但当测度存在密度函数时，我们给出充分条件以确保密度可通过该偏微分方程逐点恢复。令人惊讶的是，当空间维度为奇数时重建过程具有局部性质，而在偶数维度中则呈现非局部性。我们分别给出二维和三维空间中的重建实例，并利用所得结果刻画深度等值线的正则性。论文最后给出偶数维度非局部性的部分对偶结论。