Squeezing Constants of Bounded Homogeneous Domains

Definitions and mathematical statement

In the usual finite-dimensional formulation, let $n\geq 1$ and let $D\subset{\mathbb{C}}^n$ be a bounded domain, meaning a nonempty connected open set. Write $\mathbb B^n=\{z\in{\mathbb{C}}^n:\|z\|_2<1\}$. For $p\in D$, define the squeezing function by \[ s_D(p)=\sup\left\{r\in(0,1]:\text{there exists an injective holomorphic map }f:D\to\mathbb B^n,\ f(p)=0,\ r\mathbb B^n\subseteq f(D)\right\}. \] The domain $D$ is homogeneous if its group $\operatorname{Aut}(D)$ of biholomorphic self-maps acts transitively: for every $p,q\in D$, there exists $\phi\in\operatorname{Aut}(D)$ with $\phi(p)=q$. Biholomorphic invariance of the squeezing function then makes $s_D$ constant; denote its value by $s(D)$.

Problem. Determine $s(D)$ for every bounded homogeneous domain $D\subset{\mathbb{C}}^n$, with $n\geq1$ arbitrary.

Unresolved scope qualifications. The supplied question does not explicitly restrict the ambient dimension. The definition above states its conventional finite-dimensional interpretation. The supplied background also mentions infinite-dimensional finite-rank bounded symmetric factors, but specifies neither an ambient space nor a squeezing-function definition for a general infinite-dimensional formulation. Thus an intended extension to infinite-dimensional bounded homogeneous domains remains unspecified. The word "compute" is also not accompanied by a required form of formula or effective procedure.

Short English statement

For each bounded homogeneous complex domain, determine the common value of its squeezing function, which measures how large a centered Euclidean ball can fit inside an injective holomorphic image of the domain in the unit ball. The displayed formulation uses finite-dimensional domains; any intended infinite-dimensional scope requires clarification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003927$, source problem code catalogue entry, title "Squeezing Constants of Bounded Homogeneous Domains." The supplied record lists proposed year $2018$ and names no individual proposer.
• Supplied report citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. https://doi.org/10.4171/owr/2018/37. The literature-source label is "Oberwolfach Report 37/2018." The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Infinite dimensional holomorphic homogeneous regular domains. https://arxiv.org/abs/1808.03416. The supplied assessment attributes to Chu, Kim, and Kim the computation of squeezing constants for all bounded symmetric domains, including infinite-dimensional finite-rank factors, and states that this does not settle the broader question for arbitrary bounded homogeneous domains.
• Literature assessment date: 2026-08-21. The label "partially solved" and the preceding assessment are supplied dated curation derived from the OWR record, not independently verified here. The record also labels its statement corrected_verified and reports that the official report was checked directly; that verification claim is supplied provenance, not a source visit or independent verification by this draft.