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.

中文翻译：

---

关于多胞体角度估值连续性的可扩展性

P. McMullen 的经典定理描述了在平移下不变且弱连续的多面体的所有估值，即，相对于多面体面的平行位移是连续的。虽然通常不难检查估值是否是弱连续的，但不清楚如何决定它是否允许对凸体的连续扩展。在 McMullen 构造的一个特例中，获得了这种扩展的初始数据的一个简单的必要和充分条件。