  We study the well-posedness of a class of nonlocal-interaction equations on
general domains $\Omega\subset \mathbb{R}^d$, including nonconvex ones. We show
that under mild assumptions on the regularity of domains (uniform
prox-regularity), for $\lambda$-geodesically convex interaction and external
potentials, the nonlocal-interaction equations have unique weak measure
solutions. Moreover, we show quantitative estimates on the stability of
solutions which quantify the interplay of the geometry of the domain and the
convexity of the energy. We use these results to investigate on which domains
and for which potentials the solutions aggregate to a single point as time goes
to infinity. Our approach is based on the theory of gradient flows in spaces of
probability measures.
