Hadwiger-type characterization for locally bounded or non-negative valuations on polytopes

Establish whether every locally bounded or non-negative, translation- and rotation-invariant valuation on the space of convex polytopes in R^n is a linear combination of the intrinsic volumes.

Background

The paper proves that every measurable, translation- and SO(n)-invariant valuation on convex polytopes is a linear combination of the intrinsic volumes, thereby answering the corresponding question for measurable valuations. The cited Problem 15.4 of McMullen and Schneider poses related classification questions under weaker or different regularity assumptions.

The authors explicitly identify the cases of locally bounded valuations and non-negative valuations as unresolved. These cases are not covered by the paper's measurability-based theorem, so it remains unknown whether the same Hadwiger-type characterization holds for either class.