Minkowski Endomorphisms
Minkowski Endomorphisms
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闵可夫斯基自同态
DOI：
10.1007/s00039-017-0405-z
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发表时间：
2016
影响因子：
2.2
通讯作者：
Felix Dorrek
中科院分区：
文献类型：
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作者：
Felix Dorrek
解决了关于Minkowski自同态和Minkowski赋值的几个公开问题。更确切地说，它证明了所有的Minkowski自同态是一致连续的，但存在Minkowski自同态不是弱单调的。这回答了Kiderlen（Trans Am Math Soc 358：5539-5564，2006）、Schneider（Convex bodies：the Brunn-Minkowski theory.数学及其应用百科全书第二扩充版。剑桥大学出版社，剑桥，2014）和Schuster（Trans Am Math Soc 359：5567-5591，2007）。此外，最近的代表性结果的Minkowski估值Schuster和Wannerer的改进下额外的同质性假设。还回答了同一作者的闵可夫斯基自同态的结构有关的问题。最后，证明了在连续的、偶数的、SO（n）-等变的和平移不变的Minkowski赋值类中不存在McMullen分解，扩展了Parapatits和Wannerer的结果（杜克数学J 162：1895-1922，2013）。
Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not weakly-monotone. This answers questions posed repeatedly by Kiderlen (Trans Am Math Soc 358:5539–5564, 2006), Schneider (Convex bodies: the Brunn–Minkowski theory. Second expanded edition, encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge, 2014) and Schuster (Trans Am Math Soc 359:5567–5591, 2007). Furthermore, a recent representation result for Minkowski valuations by Schuster and Wannerer is improved under additional homogeneity assumptions. Also a question related to the structure of Minkowski endomorphisms by the same authors is answered. Finally, it is shown that there exists no McMullen decomposition in the class of continuous, even, SO(n)-equivariant and translation invariant Minkowski valuations extending a result by Parapatits and Wannerer (Duke Math J 162:1895–1922, 2013).