Name of mean- or median-like values? Consider the data points $x_i \in \mathbb R$ for $i=1,\ldots,n$ as well as following definition:
$$\hat x_c := \underset{{r\in \mathbb R}}{\operatorname{argmin}}\sum_i \vert r - x_i \vert^c $$
This definition includes the median for $c=1$ as well as the mean for $c=2$.
Is there a name for this general class of mean/median/estimators?
The following image shows an example of some data and $\hat x_c$ for various values of $c$. The red line shows the values of $\hat x_c$ (horizontal axis) depending on the values of $c$ (red vertical axis).

You can observe that for $c\to \infty$ the value of $\hat x_c$ will be more and more influenced by the outliers and converge to the mid-range $\frac{\min x_i + \max x_i }{2}$. (Which intuitively makes sense when you compare it to the behaviour of $p$-norms.) You can also generalize this simple definition to multiple dimensions ($x_i \in \mathbb R^d$) by replacing the absolute value $\vert \cdot \vert$ by a suitable norm $\Vert \cdot \Vert$.
 A: This is only a partial answer for $c \in (0,1]$:
A Fréchet mean has the form
$$\Psi(x) = \operatorname{argmin}_r \sum_i d(r, x_i)$$
where $(M,d)$ is a metric space and $x \in M^k$ and $d$ is a metric. In our  case
$$ d(x, y) = | x - y |^c$$
is indeed a metric on $M = \mathbb R$, but only for $c \in (0,1]$. In this case we can indeed call it a Fréchet mean with the given metric $d$.
A: The term "$L_p$-estimator" (i.e., L1, L2,..., where the $p$ in the literature is your $c$) is sometimes used to refer to such estimators. https://iopscience.iop.org/article/10.1088/0026-1394/43/3/004
However, I have hardly seen this term used for estimators of 1-dimensional location, more in regression (of which 1-d location is a special case).
More generally, M-estimators are defined by $\sum_{i=1}^n \rho(x_i,\theta)=\min!$.
A: A rather old paper (1917) Algebraic means by Fujisawa refers to them as power means although you have to take the $c$th root to get that. His article also discusses algebraic means which are related.
