  A Fr\'echet mean of a random variable $Y$ with values in a metric space
$(\mathcal Q, d)$ is an element of the metric space that minimizes $q \mapsto
\mathbb E[d(Y,q)^2]$. This minimizer may be non-unique. We study strong laws of
large numbers for sets of generalized Fr\'echet means. Following
generalizations are considered: the minimizers of $\mathbb E[d(Y, q)^\alpha]$
for $\alpha > 0$, the minimizers of $\mathbb E[H(d(Y, q))]$ for integrals $H$
of non-decreasing functions, and the minimizers of $\mathbb E[\mathfrak c(Y,
q)]$ for a quite unrestricted class of cost functions $\mathfrak c$. We show
convergence of empirical versions of these sets in outer limit and in one-sided
Hausdorff distance. The derived results require only minimal assumptions.
