Splitting theorem and localization for RCD(0,∞) and negative-dimension cases

Establish an isometric splitting theorem and a localization (needle decomposition) for RCD(0,∞) spaces, and for RCD(0,N) spaces with N in (−∞,−1), to enable extensions of the generalized Grünbaum inequality beyond the smooth weighted Riemannian setting.

Background

The main results rely crucially on Gigli’s splitting theorem and Cavalletti–Mondino’s localization, both currently available in RCD(0,N) with N in (1,∞). To treat N=∞ and N<−1 in the synthetic framework, analogous splitting and localization results are needed.

In the smooth setting (weighted Riemannian manifolds), splitting and localization are available for N=∞ and N<−1, but for RCD spaces these results are not yet established, limiting generalization of the paper’s inequalities.

References

However, both the splitting theorem and localization are not known even for N = ∞, thereby we need to generalize them or consider a different method.

— A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces (2408.15030 - Brunel et al., 27 Aug 2024) in Section 7 (Further problems), item (A)