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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 Euclideandistance between $x_1$ and $x_2$. We show, if $\rho(x)$ is a mixture ofGaussians whose covariance matrix is the identity, and if the means of theGaussians are in generic position, then the density $\rho(x)$ isreconstructible, 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 berelated by a rigid motion $g\in E(d)$ as $\rho(x)=\bar{\rho} (g\cdot x)$. Wealso show that a similar result holds when the distance is defined by asymmetric bilinear form.

The shape of a Gaussian mixture is characterized by the probability density of the distance between two samples