Full characterization of Val_j(R^n) for 1 ≤ j ≤ n−2

Establish a complete characterization of the j-homogeneous components Val_j(R^n) of the space of continuous and translation-invariant valuations on convex bodies in R^n for indices 1 ≤ j ≤ n−2, beyond the already understood cases j=0 (Euler characteristic), j=n (Lebesgue measure), and j=n−1 (area-measure integral representation).

Background

The paper studies zonal valuations—those invariant under SO(n−1)—and provides a complete classification for these, including principal value integral representations against area measures. In contrast, for the full space Val(R^n), McMullen’s decomposition identifies homogeneous components Val_j(R^n), and classical results describe Val_0, Val_n, and Val_{n−1}. However, outside these cases, the general structural description of Val_j(R^n) has been a longstanding gap.

Within the Introduction, the authors explicitly note that a full characterization of Val_j(R^n) for 1 ≤ j ≤ n−2 is not known. Their results address the zonal subspace Val_j(R^n)^{SO(n−1)} but do not resolve the broader classification problem for Val_j(R^n) without symmetry assumptions.