Fréchet mean

Remarks

The condition on $q$ and the triangle inequality give $F(p) < \infty$ for every $p$. The set of Fréchet means may be empty or have several points. On a complete Riemannian manifold a local minimiser of $F$ is called a Karcher mean. Replacing $d^2$ by $d$ gives the Fréchet median.

Example: in $M = \R^k$ with the Euclidean distance, $\sum_i \norm{x_i - p}^2 = \sum_i \norm{x_i - \bar x}^2 + n \norm{\bar x - p}^2$, where $\bar x$ is the sample mean. So the sample Fréchet mean is unique and equals $\bar x$.

Non-uniqueness: on the unit sphere with the great-circle distance, take two antipodal points $x$ and $-x$. If $p$ is at distance $\theta$ from $x$, it is at distance $\pi - \theta$ from $-x$, and $\theta^2 + (\pi - \theta)^2$ is smallest exactly at $\theta = \pi/2$. So every point of the equator between them is a Fréchet mean, and the "midpoint" is not a single point.

Non-example: if $X$ is real with the Cauchy distribution, then $E[(X - q)^2] = \infty$ for every $q \in \R$. The moment condition fails, $F$ is infinite everywhere, and $X$ has no Fréchet mean, just as it has no ordinary mean.