Do barycenters of convex sets lie inside the set in RCD(0,N) spaces?

Determine whether, in an RCD(0,N) metric measure space with N in (1,∞), for any convex set Ω with finite measure, every barycenter of the uniform probability measure on Ω belongs to Ω (i.e., lies inside the convex set).

Background

The paper generalizes Grünbaum’s inequality to RCD(0,N) spaces and discusses barycenters defined via minimization of second-moment-type functionals for probability measures. For uniform measures on convex sets in Euclidean spaces, the barycenter (centroid) lies in the set, but in RCD(0,N) spaces barycenters may be non-unique.

The authors provide an example (a cylinder R × S^1 with Ω = [-1,1] × S^1) where many barycenters exist, all lying on {0} × S^1, and note uncertainty about whether barycenters always lie in Ω in general.