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Let $(M,g)$ be a Riemannian manifold. If $\mu$ is a probability measure on$M$ given by a continuous density function, one would expect the Fr\'{e}chetmeans of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $\mu$,to behave ``generically''; e.g. the probability that the Fr\'{e}chet mean set$\mbox{FM}(Q)$ has any elements that lie in a given, positive-codimensionsubmanifold, should be zero for any $N\geq 1$. Even this simplest instance ofgenericity does not seem to have been proven in the literature, except inspecial cases. The main result of this paper is a general, and stronger,genericity property: given i.i.d. absolutely continuous $M$-valued randomvariables $X_1,\dots, X_N$, and a subset $A\subset M$ of volume-measure zero,$\mbox{Pr}\left\{\mbox{FM}(\{X_1,\dots,X_N\})\subset M\backslash A\right\}=1.$We also establish a companion theorem for equivariant Fr\'{e}chet means,defined when $(M,g)$ arises as the quotient of a Riemannian manifold$(\widetilde{M},\tilde{g})$ by a free, isometric action of a finite group. Theequivariant Fr\'{e}chet means lie in $\widetilde{M}$, but, as we show, projectdown to the ordinary Fr\'{e}chet sample means, and enjoy a similar genericityproperty. Both these theorems are proven as consequences of a purely geometric(and quite general) result that constitutes the core mathematics in this paper:If $A\subset M$ has volume zero in $M$ , then the set $\{Q\in M^N :\mbox{FM}(Q) \cap A\neq\emptyset\}$ has volume zero in $M^N$. We conclude thepaper with an application to partial scaling-rotation means, a type of mean forsymmetric positive-definite matrices.

A genericity property of Fr\'{e}chet sample means on Riemannian manifolds