Let $X$ be a random variable on $\mathbb R^n$ with density $f$ (w.r.t Lebesgue).
Question. What is a generally agreed upon definition of unimodularity (of the distribution) of $X$ ? Are there any known charaterizations of this property ?
Notes
In the special case $n=1$, unimodality is agreed in the literature to mean that
There is a point $c \in \mathbb R$ (called the mode of $X$) such that $f$ is increasing on $(-\infty, c)$ and decreasing on $(c,\infty)$.
In this case, it is well-known (Khintchine, 1938) that $X$ has unimodal distribution iff $X \overset{d}{=} UZ$, where $U$ is the uniformly distributed r.v on $[0, 1]$ and $Z$ a r.v on $\mathbb R$ which is independent of $U$.
Update: $\alpha$-unimodularity
The following definition is an interesting candidate for the concept of multivariate modularity.
Definition. Let $\alpha \in [1,n]$. A random variable $X$ on $\mathbb R^d$ is said to $\alpha$-unimodular about the point $c \in \mathbb R^d$ iff for every bounded measurable function $g:\mathbb R^n \rightarrow [0,\infty)$ the function $t \mapsto t^{n-\alpha}\mathbb E[g(t(X-c))]$ is non-decreasing on $[0,\infty]$. The unimodularity index of $X$ is defined to be the smallest value of $\alpha \in [1,n]$ such that $X$ is $\alpha$-unimodular.
The following is a characterization of multivariate unimodularity which parallels the univariate case.
Theorem (Dharmadhikari et Joag-Dev, 1988). $X$ is $\alpha$-unimodular iff $X \overset{d}{=} U^{1/\alpha}Z$, where $U$ is the uniformly distributed r.v on $[0, 1]$ and $Z$ a r.v on $\mathbb R^d$ which is independent of $U$.