凸体的卷积与乘子变换
Convolutions and multiplier transformations of convex bodies
摘要：
研究了Rn中凸体集到自身的旋转交织映射，这些映射是关于Minkowski和Blaschke加法的连续线性算子.主要关注的是Blaschke-Minkowski同态。我们证明了这样的映射是由一个球面卷积算子表示的。这种表示的一个应用是一个完整的分类，所有甚至Blaschke-Minkowski同态表明，这些地图的行为在许多方面类似于著名的投影体算子。在进一步的应用中有以下结果：如果一个偶Blaschke-Minkowski同态映射一个凸体到一个多面体，那么它是投影体算子的常数倍。
英文摘要：
Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.
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