Is there an accepted definition for the median of a sample on the plane, or higher ordered spaces? If so, what?
If not, why not?
For a sample on the line, the median minimizes the total absolute deviation. It would seem natural to extend the definition to R2, etc., but I've never seen it. But then, I've been out in left field for a long time.
 A: The Tukey halfspace median can be extended to >2 dimensions using DEEPLOC, an algorithm due to Struyf and Rousseeuw; see here for details.
The algorithm is used to approximate the point of greatest depth efficiently; naive methods which attempt to determine this exactly usually run afoul of (the computational version of) "the curse of dimensionality", where the runtime required to calculate a statistic grows exponentially with the number of dimensions of the space.
A: A definition that comes close to it, for unimodal distributions, is the tukey halfspace median 


*

*http://cgm.cs.mcgill.ca/~athens/Geometric-Estimators/halfspace.html

*http://www.isical.ac.in/~statmath/html/publication/Tukey_tech_rep.pdf

*https://www.isical.ac.in/~statmath/report/11310-15.pdf
A: I'm not sure there is one accepted definition for a multivariate median.  The one I'm familiar with is Oja's median point, which minimizes the sum of volumes of simplices formed over subsets of points.  (See the link for a technical definition.)
Update: The site referenced for the Oja definition above also has a nice paper covering a number of definitions of a multivariate median:


*

*Geometric Measures of Data Depth
A: As @Ars said there are no accepted definition (and this is a good point). There are general alternatives families of ways to generalize quantiles on $\mathbb{R}^d$, I think the most significant are:

*

*Generalize quantile process Let $P_n(A)$ be the empirical measure (=the proportion of observations in $A$). Then, with $\mathbb{A}$ a well chosen subset of the Borel sets in $\mathbb{R}^d$ and $\lambda$ a real valued measure,
you can define the empirical quantile function:

$U_n(t)=\inf (\lambda(A) : P_n(A)\geq t A\in\mathbb{A})$
Suppose you can find one $A_{t}$ that gives you the minimum. Then the set (or an element of the set) $A_{1/2-\epsilon}\cap A_{1/2+\epsilon}$ gives you the median when $\epsilon$ is made small enough. The definition of the median is recovered when using $\mathbb{A}=(]-\infty,x] x\in\mathbb{R})$  and $\lambda(]-\infty,x])=x$. Ars answer falls into that framework I guess...  tukey's half space location may be obtained using $\mathbb{A}(a)=( H_{x}=(t\in \mathbb{R}^d :\; \langle a, t \rangle \leq x ) $ and  $\lambda(H_{x})=x$  (with $x\in \mathbb{R}$, $a\in\mathbb{R}^d$).

*

*variational definition and M-estimation
The idea here is that the  $\alpha$-quantile $Q_{\alpha}$ of a random variable $Y$ in $\mathbb{R}$ can be defined through a variational equality.


*The most common definition is using the quantile regression function $\rho_{\alpha}$ (also known as pinball loss, guess why ? )  $Q_{\alpha}=arg\inf_{x\in \mathbb{R}}\mathbb{E}[\rho_{\alpha}(Y-x)]$. The case $\alpha=1/2$ gives $\rho_{1/2}(y)=|y|$ and you can generalize that to higher dimension using $l^1$ distances as done in @Srikant Answer. This is theoretical median but gives you empirical median if you replace expectation by empirical expectation (mean).


*But Kolshinskii proposes to use  Legendre-Fenchel transform: since $Q_{\alpha}=Arg\sup_s (s\alpha-f(s))$
where $f(s)=\frac{1}{2}\mathbb{E} [|s-Y|-|Y|+s]$ for $s\in \mathbb{R}$.
He gives a lot of deep reasons for that (see the paper ;)). Generalizing this to higher dimensions require working with a vectorial $\alpha$ and replacing $s\alpha$ by $\langle s,\alpha\rangle$ but you can take $\alpha=(1/2,\dots,1/2)$.


*Partial ordering You can generalize the definition of quantiles in $\mathbb{R}^d$ as soon as you can create a partial order (with equivalence classes).
Obviously there are bridges between the different formulations. They are not all obvious...
A: There are distinct ways to generalize the concept of median to higher dimensions.  One not yet mentioned, but which was proposed long ago, is to construct a convex hull, peel it away, and iterate for as long as you can: what's left in the last hull is a set of points that are all candidates to be "medians."
"Head-banging" is another more recent attempt (c. 1980) to construct a robust center to a 2D point cloud.  (The link is to documentation and software available at the US National Cancer Institute.)
The principal reason why there are multiple distinct generalizations and no one obvious solution is that R1 can be ordered but R2, R3, ... cannot be.
A: Geometric median is the point with the smallest average euclidian distance from the samples
A: I do not know if any such definition exists but I will try and extend the standard definition of the median to $R^2$. I will use the following notation:
$X$, $Y$: the random variables associated with the two dimensions.
$m_x$, $m_y$: the corresponding medians.
$f(x,y)$: the joint pdf for our random variables
To extend the definition of the median to $R^2$, we choose $m_x$ and $m_y$ to minimize the following:
$E(|(x,y) - (m_x,m_y)|$
The problem now is that we need a definition for what we mean by:
$|(x,y) - (m_x,m_y)|$
The above is in a sense a distance metric and several possible candidate definitions are possible. 
Eucliedan Metric
$|(x,y) - (m_x,m_y)| = \sqrt{(x-m_x)^2 + (y-m_y)^2}$ 
Computing the median under the euclidean metric will require computing the expectation of the above with respect to the joint density $f(x,y)$.
Taxicab Metric
$|(x,y) - (m_x,m_y)| = |x-m_x| + |y-m_y|$ 
Computing the median in the case of the taxicab metric involves computing the median of $X$ and $Y$ separately as the metric is separable in $x$ and $y$. 
