Holomorphic Homogeneous Regularity of Infinite-Dimensional Convex Domains

Definitions and mathematical statement

Let $H$ be an infinite-dimensional complex Hilbert space with norm $\|\cdot\|$, and write $B_H=\{z\in H:\|z\|<1\}$. A domain is a nonempty connected open subset of $H$. A domain $D\subset H$ is bounded if $\sup_{z\in D}\|z\|<\infty$, and convex if $(1-t)x+ty\in D$ for all $x,y\in D$ and $t\in[0,1]$.

For a precise formulation, adopt the following Hilbert-ball convention. Holomorphic maps are complex Fréchet-holomorphic maps. A biholomorphic embedding $f:D\to B_H$ means a biholomorphism from $D$ onto an open subset $f(D)\subset B_H$. For $p\in D$, define the squeezing function by \[ s_D(p)=\sup\left(\{0\}\cup\left\{r\in(0,1]:\text{there exists a biholomorphic embedding }f:D\to B_H, \ f(p)=0,\ rB_H\subset f(D)\right\}\right). \] The domain $D$ is called holomorphic homogeneous regular (HHR) if \[ \inf_{p\in D}s_D(p)>0. \] Equivalently, there exists $r_D\in(0,1)$ such that for every $p\in D$ there is such an embedding $f_p$ satisfying \[ r_DB_H\subset f_p(D)\subset B_H,\qquad f_p(p)=0. \] The radius may depend on $D$, but must be independent of $p$. This definition does not require a single radius valid for all domains.

Principal question. Is every bounded convex domain $D$ in every infinite-dimensional complex Hilbert space $H$ HHR? In the above convention, is \[ \forall H\ \forall D\subset H\text{ bounded convex domain},\qquad \inf_{p\in D}s_D(p)>0 \] true, where $H$ ranges over infinite-dimensional complex Hilbert spaces?

Additional question retained from the original statement. A domain is homogeneous if its group $\operatorname{Aut}(D)$ of biholomorphic self-maps acts transitively on $D$. Under the preceding squeezing convention, $s_D$ is invariant under $\operatorname{Aut}(D)$, so for a homogeneous domain it has a constant value $c_D$. The original statement additionally asks whether one can compute this constant for bounded homogeneous domains. It does not specify a family of domains, the desired form of the computation, or whether this additional question has the same infinite-dimensional Hilbert-space scope. Its full intended target therefore remains unresolved.

Formulation qualifications. The supplied record does not explicitly define the infinite-dimensional squeezing function or HHR, and does not state a separability restriction. The Hilbert-ball and biholomorphic-embedding conventions above require comparison with the cited source; the unrestricted quantifier over $H$ reflects the absence of a stated separability hypothesis. The supplied clean statement retains only the principal question, whereas the original statement also contains the computation question. The supplied literature assessment refers separately to bounded symmetric domains; this does not identify the entire class of bounded homogeneous domains with bounded symmetric domains.

Short English statement

For each bounded convex domain in an infinite-dimensional complex Hilbert space, must there be a positive radius, independent of the chosen point, such that the domain can be mapped biholomorphically into the unit ball with that point at the origin and with the image containing the ball of that radius? The original record also asks for the constant squeezing value of bounded homogeneous domains, with its precise scope unspecified. The supplied status is open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003926$, title Holomorphic Homogeneous Regularity of Infinite-Dimensional Convex Domains, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns proposed year 2018 and names no proposer.
• Supplied source citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/37, pp. 2253--2302. Supplied source URL and literature-source link labelled Oberwolfach Report 37/2018: https://doi.org/10.4171/owr/2018/37. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature-source link: Infinite dimensional holomorphic homogeneous regular domains, https://arxiv.org/abs/1808.03416. No authors or further bibliographic details are supplied in the record.
• Literature assessment dated 2026-08-21: the supplied curation labels the convex infinite-dimensional Hilbert-space HHR question open, says that later work proves HHR only for uniformly elliptic domains, and says that the bounded-symmetric-domain part is addressed separately. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official Oberwolfach report was checked directly and that the clean statement preserves the target while restoring definitions and removing source-dependent wording. These are claims of the supplied record; no source visit or independent verification was performed for this draft.