Radius and Fixed Points of Convex Building Subsets

Definitions and mathematical statement

Let $(B,d_B)$ be a spherical building with its standard spherical metric, normalized so that apartments are unit spherical Coxeter complexes. For a subset $C\subseteq B$, use the restricted metric $d_C=d_B|_{C\times C}$. Interpret convexity as $\pi$-convexity: every geodesic segment in $B$ of length strictly less than $\pi$ whose endpoints lie in $C$ is contained in $C$. Closedness is with respect to $d_B$. Here a subbuilding means a subset carrying a spherical building structure inherited from $B$; the supplied record does not specify the precise convention for inherited apartments or building type.

For nonempty $C$, define its intrinsic circumradius by \[ \operatorname{rad}_C(C):=\inf_{x\in C}\sup_{y\in C}d_C(x,y). \] Let \[ \operatorname{Isom}(C):=\{g:C\to C\mid g\text{ is bijective and }d_C(gx,gy)=d_C(x,y)\text{ for all }x,y\in C\}. \] A fixed point of this group means a common fixed point in $C$.

Under these conventions, the supplied question is whether, for every spherical building $B$ and every nonempty closed convex subset $C\subseteq B$ that is not a subbuilding, both assertions hold: \[ \operatorname{rad}_C(C)\leq\frac{\pi}{2}, \qquad \exists c\in C\ \forall g\in\operatorname{Isom}(C),\quad g(c)=c. \] The record does not explicitly assume nonemptiness or define $\operatorname{rad}(C)$. Thus this is a conditional precise formulation using the nonempty, intrinsic-radius convention. If centers are instead allowed in $B$, the relevant quantity is \[ \operatorname{rad}_B(C):=\inf_{x\in B}\sup_{y\in C}d_B(x,y), \] and the intended radius assertion must be confirmed. The record supplies no thickness, dimension, rank, or building-type restriction, and does not require $C$ to be a simplicial subcomplex. These unresolved conventions must be checked before treating this draft as a definitive formulation.

Short English statement

For a nonempty closed convex subset of a spherical building that is not itself a subbuilding, ask whether its intrinsic circumradius is at most $\pi/2$ and whether all its isometries fix a common point in the subset. The radius convention, nonemptiness, and precise subbuilding convention require confirmation. The supplied record labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record registry; supplied permanent ID $30001311$, exact source code catalogue entry, title Radius and Fixed Points of Convex Building Subsets. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2009 and names no individual proposer.
• Supplied report citation: Differentialgeometrie im Großen (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/35, pp. 1943--1972. https://doi.org/10.4171/owr/2009/35. The background attributes the problem to workshop open-problem discussions documented by Oberwolfach Reports.
• Supplied literature reference, labeled "Ramos-Cuevas theorem": https://arxiv.org/abs/1408.3017.
• Supplied literature reference, labeled "Kleiner--Leeb rigidity questions": https://math.nyu.edu/~bkleiner/convex.pdf.
• Supplied literature assessment checked at 2026-08-21: status "partially solved." The assessment states that the radius and fixed-point assertions are proved in several cases, including low dimensions, type A, type D, and the thick-building center conjecture, while the full general formulation remains unsettled. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The supplied statement-status label is corrected_verified. Its verification note says that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification claim, not verification performed for this draft.