We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.

中文翻译：

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从矩重建凸体

我们研究了从有限数量的几何矩中可以检索到多少关于凸体的信息。我们给出了凸体由其有限数量的几何矩唯一确定的充分条件，并且我们证明在所有凸体中，由有限数量的矩唯一确定的那些形成稠密集。此外，我们基于几何矩推导出凸体的稳定性结果。事实证明，通过使用另一组矩，即勒让德矩，稳定性结果得到了显着改善。我们提出了一种重建算法，该算法使用有限数量的勒让德矩来近似凸体。使用勒让德矩的稳定性结果建立算法的一致性。当只有勒让德矩的噪声测量可用时，