Rigidity of Large-Radius Convex Subsets in Buildings

Definitions and mathematical statement

Let $B$ be a spherical building equipped with its standard angular metric $d_B$; its apartments are unit spherical Coxeter complexes. A nonempty subset $C\subseteq B$ is called $\pi$-convex if, for every $x,y\in C$ with $d_B(x,y)<\pi$, the unique geodesic segment from $x$ to $y$ lies in $C$. No condition on geodesics joining points at distance exactly $\pi$ is imposed by this definition.

Two possible circumradius conventions are \[ \operatorname{rad}_{C}(C)=\inf_{c\in C}\sup_{x\in C}d_B(c,x), \qquad \operatorname{rad}_{B}(C)=\inf_{b\in B}\sup_{x\in C}d_B(b,x). \] The first restricts centers to $C$; the second allows centers anywhere in $B$. Neither definition requires the infimum to be attained.

Source question, with unresolved conventions. Are convex subsets of spherical buildings with circumradius strictly greater than $\pi/2$ necessarily subbuildings? Here "rigid" means precisely the asserted subbuilding conclusion, rather than an additional rigidity property.

Under the explicit conventions that convexity means $\pi$-convexity and circumradius means $\operatorname{rad}_{C}$, the proposed universal assertion takes the form \[ \begin{gathered} \text{For every spherical building }B\text{ and every nonempty }C\subseteq B,\\ C\text{ is }\pi\text{-convex and }\operatorname{rad}_{C}(C)>\frac{\pi}{2} \quad\Longrightarrow\quad C\text{ is a subbuilding of }B. \end{gathered} \] This is a conditional transcription of the intended question, not a resolution of its missing conventions. If ambient centers are intended, replace $\operatorname{rad}_{C}$ by $\operatorname{rad}_{B}$. The strict inequality excludes circumradius exactly $\pi/2$.

A subbuilding is understood informally here as a subset carrying a spherical-building structure compatible with that of $B$. The supplied record does not specify the permitted rank, type, or precise compatibility requirement for this conclusion. It also does not specify closedness of $C$, thickness or rank restrictions on $B$, or whether its use of "convex" includes a condition at antipodal pairs. These qualifications require clarification before the displayed assertion can be treated as a fully determined formalization. The supplied target concerns arbitrary convex subsets; it does not restrict them to convex subcomplexes.

Short English statement

The question asks whether every convex subset of a spherical building whose circumradius is strictly greater than $\pi/2$ must itself be a subbuilding. The radius convention, precise convexity convention, and intended meaning of subbuilding need clarification. The supplied record labels the problem partially solved and reports that the unrestricted convex-subset question remains open; these are supplied dated assessments requiring independent review.

Sources

• Local UnsolvedMath problem registry, permanent ID $30001811$, source problem code catalogue entry, title "Rigidity of Large-Radius Convex Subsets in Buildings," in the collection "Oberwolfach Reports Open Problems." The supplied proposed year is 2011; no individual proposer is supplied.
• Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. Supplied original-report link: https://doi.org/10.4171/owr/2011/30. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page is supplied.
• Supplied literature source labeled "Ramos--Cuevas convex subcomplexes": https://arxiv.org/abs/1408.3017. The record supplies no further bibliographic details for this reference.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified here. It labels the problem "partially solved" and states that the Kleiner--Leeb dichotomy is proved for convex subcomplexes in important building types, including type A and the relevant type D cases; that the center conjecture is settled for thick buildings; and that the unrestricted convex-subset question remains open. The background describes this as literature triage derived from the supplied Oberwolfach curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports that the mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; the cited sources have not been independently inspected for this draft.