Existence of the local volume limit for valuations centered along a specialization

Determine whether the limit defining the local volume invariant exists, rather than only its limit superior, for valuations in \(\mathrm{Val}^*_{X,\ni x}\) whose centers need not equal \(x\).

Background

The paper extends the local SS-invariant from valuations centered at the closed point xx to valuations whose centers specialize to xx. For this broader class, the local volume is defined using a limsup of normalized lengths. The authors explicitly identify the existence of an actual limit as unknown, which prevents them from using the more standard definition in complete generality.

References

Unlike the case of $\nu\in \mathrm{Val}_{X,x}$ (cf. Lemma-Definition 3.1), we do not know whether the limit exists in general. Hence, we use the limit supremum of real numbers.

— On the existence of minimizer on a log Fano cone singularity  (2608.23019 - Kim, 24 Aug 2026) in Section 2, subsection “Local S-invariant,” immediately after Definition \ref{yongheungdong}