  For $1 \le p < \infty$, the Fr\'echet $p$-mean of a probability distribution
$\mu$ on a metric space $(X,d)$ is the set $F_p(\mu) := {\arg\,\min}_{x\in
X}\int_{X}d^p(x,y)\, d\mu(y)$, which is taken to be empty if no minimizer
exists. Given a sequence $(Y_i)_{i \in \mathbb{N}}$ of independent, identically
distributed random samples from some probability measure $\mu$ on $X$, the
Fr\'echet $p$-means of the empirical measures,
$F_p(\frac{1}{n}\sum_{i=1}^{n}\delta_{Y_i})$ form a sequence of random closed
subsets of $X$. We investigate the senses in which this sequence of random
closed sets and related objects converge almost surely as $n \to \infty$.
