What is the medoid function and its relation to the median? 
The medoid function is defined in this  graph neural network paper as:
$$ t := \arg\min_{y\in \mathcal{X}}\sum_{j=1}^N||x_j-y||$$
which is a "multivariate generalization of the Median" and $\mathcal{X}=\{x_1,x_2,...,x_n\}$ is a collection of points in $\mathbb{R}^d$.

How does this relate to the median in higher dimensions? In $\mathbb{R}^1$, the median uses an ordered list of data-points and finds the central point, what does it mean to find the median in $\mathbb{R}^2$ and beyond, and how does this function do it?
I've never seen this function before, so i wonder if it has it's drawbacks and if there are better formulas to use? Furthermore, Wikipedia states the medoid "is not equivalent to a median" so is this paper wrong?

In one dimension, the median minimises the sum of absolute values of distances,
$$\textrm{median} = \arg\min_m \sum_{j=1}^n |x_i-m|$$
and thus is a medoid. In that sense it's a generalisation of the median, and it's a reasonable location summary. As Wikipedia goes on to say, medoids aren't equivalent to medians in general, because medians are really a one-dimensional concept.
There are  multiple generalisations of the median to higher dimensions, all trying to preserve some properties of the median, and they are all different. A geometric median is like a medoid except it doesn't have to be one of the sample points; this paper reviews some other definitions of higher-dimensional medians
