Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties?
- The distribution of $X$ is log-concave, i.e. for every $n$ the joint distribution of $(X_1,\dots,X_n)$ has a log-concave density on $\mathbb{R}^n$.
- $X$ is isotropic, i.e. $\mathsf{E} \, X_i = 0$, $\mathsf{E} \, X_i X_j = \delta_{ij}$ (in particular, for all $\xi \in \ell^2$ the "scalar product" $\langle X, \xi \rangle$ is a well-defined random variable with variance $\Vert \xi \Vert^2$).
- $X$ is determined by any of its codimension $1$ projections, i.e. for every closed codimension $1$ subspace $H \subset \ell^2$ $X$ is measurable with respect to the $\sigma$-algebra generated by $\{\langle X, \xi \rangle, \xi \in H \}$.
An equivalent formulation of the last property is:
- For every $\xi \in \ell^2 \setminus \{0\}$ the distribution of the shifted random vector $\xi + X$ is singular to that of $X$.
In connection to the last one I'd like to point out that there are isotropic log-concave random vectors whose "space of admissible shifts" is smaller than $\ell^2$ (example: $X_i$ are i.i.d. uniform on $[-1,1]$ - that has $\ell^1$ as its space of shifts). The question is: can this space be $\{0\}$?
P.S. Intuitively, this question is related to my other question: How large are the smallest-area projections of a high-dimensional convex body?, but I'm not sure how to relate them rigorously...
