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This is a question about marginals of probability measures, which seems unrelated to previous questions.

Let $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ be the unit sphere. Assume that for each $\theta\in \mathbb{S}^{d-1}$ there is an associated probability measure $\mu_\theta$ on $\mathbb{R}$.

Question: Under what conditions does there exist a probability measure $\mu$ on $\mathbb{R}^d$ such that

$$\mbox{if }X\sim \mu,\mbox{ then }\langle \theta,X\rangle \sim \mu_\theta\mbox{ for all }\theta\in \mathbb{S}^{d-1}.$$

(Here $\langle\cdot,\cdot\cdot\rangle$ is the Euclidean inner product and $A\sim \nu$ means object $A$ has prob. law $\nu$.)

A necessary condition is that the map $\theta\mapsto \mu_\theta$ be continuous under the weak topology on probability measures on $\mathbb{R}$. Is this condition sufficient?

(Motivation comes from the study of the Sliced Wasserstein distance. If the answer to the above question is "yes", then in principle one can "easily compute" barycenters for SW.)

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$\newcommand\th\theta\newcommand\vpi\varphi\newcommand\R{\mathbb R}\newcommand\S{\mathbb S^{d-1}}$Condensed version of the answer: If $\th\cdot X\sim\mu_\th$ for all $\th\in\S$, then for each nonzero $t\in\R^d$ the distribution, say $\mu_t$, of $t\cdot X$ is determined by a simple rescaling of $\mu_\th$ (the case $t=0$ is trivial). On the other hand, by Bochner's theorem, a family $(\mu_t)_{t\in\R^d}$ of probability measures on $\R$ is the family of the distributions of the random variables $t\cdot X$ (for some random vector $X$ in $\R^d$) iff the function $\R^d\ni t\mapsto\vpi(t):=\int_{\R}e^{ix}\mu_t(dx)$ is positive definite and continuous. Thus, for any given family $(\mu_\th)_{\th\in\S}$ of probability measures on $\R$, your desired condition in question will hold if and only if $\vpi$ is positive definite and continuous.

Detailed version of the answer: Suppose that $$\text{there is a probability measure $\mu$ on $\R^d$ such that},\\ \text{if $X\sim\mu$, then $\th\cdot X\sim\mu_\th$ for all $\th\in\S$.}\tag{0}$$ Then $$E_\mu f(\th\cdot X)=\int_{\R}f\,d\mu_\th \tag{1}$$ for all bounded continuous functions $f\colon\R\to\R$, where $E_\mu$ denotes the expectation assuming that $X\sim\mu$. In particular, (1) implies $$E_\mu e^{ir\th\cdot X}=\int_{\R}e^{irx}\,\mu_\th(dx) \tag{2}$$ for all $r\in\R$ and all $\th\in\S$, so that $$E_\mu e^{it\cdot X}=\vpi(t)$$ for all $t\in\R^d$, where $$\vpi(t):= \left\{ \begin{aligned} \int_{\R}e^{i|t|x}\,\mu_{t/|t|}(dx) &\text{ if }t\in\R^d\setminus\{0\}, \\ 1 &\text{ if }t=0. \end{aligned} \right. \tag{3}$$ So, if the condition (0) holds, then the function $\vpi$ -- being the characteristic function (c.f.) of $X$ -- is positive definite and continuous.

Vice versa, by Bochner's theorem, if $\vpi$ is positive definite and continuous, then there is a probability measure $\mu$ on $\R^d$ such that $E_\mu e^{it\cdot X}=\vpi(t)$ for all $t\in\R^d$, so that (2) holds for all $r\in\R$ and all $\th\in\S$, that is, the c.f. of $\th\cdot X$ is the same as the c.f. of $\mu_\th$, which means that $\th\cdot X\sim\mu_\th$. So, (0) holds if the function $\vpi$, defined by (3), is positive definite and continuous.

Thus, for any given family $(\mu_\th)_{\th\in\S}$ of probability measures on $\R$, condition (0) will hold if and only if $\vpi$ is positive definite and continuous.


Somewhat related to this is the work by Shepp and his co-authors on probabilistic tomography, about partial restoration of the distribution of a random vector $X$ in $\R^d$ based on the known distributions of a finite number of linear functionals of $X$; see e.g. this paper and references there.

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