On the extendability by continuity of angular valuations on polytopes
On the extendability by continuity of angular valuations on polytopes
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DOI：
10.1016/j.jfa.2020.108665
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发表时间：
2019-07
影响因子：
1.7
通讯作者：
Thomas Wannerer
中科院分区：
文献类型：
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作者：
Thomas Wannerer
P.McMullen的一个经典定理描述了多面体上的所有赋值，这些多面体在平移下是不变的，并且是弱连续的，即关于多面体的面片的平行位移是连续的。虽然通常不难检查估值是弱连续的，但如何确定它是否允许连续扩张到凸体并不清楚。在McMullen构造的一种特殊情况下，得到了关于这种扩张的初始数据的一个简单的充要条件。
A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult to check that a valuation is weakly continuous, it is not clear how to decide whether it admits a continuous extension to convex bodies. In a special case of McMullen's construction a simple necessary and sufficient condition on the initial data of such an extension is obtained.