For any linear transformation and two convex closed sets, we provide necessary and sufficient conditions for the transformation of the intersection of the sets to coincide with the intersection of their images. We also identify conditions for non-convex closed sets, continuous transformations, and multiple sets. We demonstrate the usefulness of our results via an application to the economics literature of mechanism design.

中文翻译：

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关于相交的线性变换

对于任何线性变换和两个凸封闭集，我们为变换集的交集以使其图像的交集提供了必要和充分的条件。我们还确定了非凸封闭集，连续变换和多个集合的条件。我们通过将机理设计应用于经济学文献，证明了我们的研究结果的实用性。