  Defining a successful notion of a multivariate quantile has been an open
problem for more than half a century, motivating a plethora of possible
solutions. Of these, the approach of [8] and [25] leading to M-quantiles, is
very appealing for its mathematical elegance combining elements of convex
analysis and probability theory. The key idea is the description of a convex
function (the K-function) whose gradient (the K-transform) is in one-to-one
correspondence between all of R^d and the unit ball in R^d. By analogy with the
d=1 case where the K-transform is a cumulative distribution function-like
object (an M-distribution), the fact that its inverse is guaranteed to exist
lends itself naturally to providing the basis for the definition of a quantile
function for all d>=1. Over the past twenty years the resulting M-quantiles
have seen applications in a variety of fields, primarily for the purpose of
detecting outliers in multidimensional spaces. In this article we prove that
for odd d>=3, it is not the gradient but a poly-Laplacian of the K-function
that is (almost everywhere) proportional to the density function. For d even
one cannot establish a differential equation connecting the K-function with the
density. These results show that usage of the K-transform for outlier detection
in higher odd-dimensions is in principle flawed, as the K-transform does not
originate from inversion of a true M-distribution. We demonstrate these
conclusions in two dimensions through examples from non-standard asymmetric
distributions. Our examples illustrate a feature of the K-transform whereby
regions in the domain with higher density map to larger volumes in the
co-domain, thereby producing a magnification effect that moves inliers closer
to the boundary of the co-domain than outliers. This feature obviously disrupts
any outlier detection mechanism that relies on the inverse K-transform.
