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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 oflarge numbers for sets of generalized Fr\'echet means. Followinggeneralizations 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 showconvergence of empirical versions of these sets in outer limit and in one-sidedHausdorff distance. The derived results require only minimal assumptions.

Strong Laws of Large Numbers for Generalizations of Fr\'echet Mean Sets