  Let $\bf{x}$ be a random variable with density $\rho(x)$ taking values in
${\mathbb R}^d$. We are interested in finding a representation for the shape of
$\rho(x)$, i.e. for the orbit $\{ \rho(g\cdot x) | g\in E(d) \}$ of $\rho$
under the Euclidean group. Let $x_1$ and $x_2$ be two random samples picked,
independently, following $\rho(x)$, and let $\Delta$ be the squared Euclidean
distance between $x_1$ and $x_2$. We show, if $\rho(x)$ is a mixture of
Gaussians whose covariance matrix is the identity, and if the means of the
Gaussians are in generic position, then the density $\rho(x)$ is
reconstructible, up to a rigid motion in $E(d)$, from the density of
$\bf{\Delta}$. In other words, any two such Gaussian mixtures $\rho(x)$ and
$\bar{\rho} (x)$ with the same distribution of distances are guaranteed to be
related by a rigid motion $g\in E(d)$ as $\rho(x)=\bar{\rho} (g\cdot x)$. We
also show that a similar result holds when the distance is defined by a
symmetric bilinear form.
