It's known that for a random vector $(X\_1,\dots,X\_n)\in \mathbb{R}^n$ with a log-concave distribution, any subvector has a long-concave distribution. I'm wondering if there are any results about its converse, in particular, when $(X\_1,\dots,X\_n)$ is isotropic, $X\_i$'s are identically distributed (but not necessarily independent) and each $X\_i$ is log-concave.


Is this true when $n=2$? That is, 



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> Suppose that $X$ and $Y$ are identically distributed subject to some log-concave distribution with $E[X]=E[Y]=0$, $E[X^2]=E[Y^2]=1$ and $E[XY]=0$. Is the joint distribution of $(X,Y)$ log-concave?
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marginal log-concave distributions and joint log-concave distributions
No. Let $X$ be, say, a standard normal random variable, $Z$ an independent random variable with $P[Z=1] = P[Z=-1] = 1/2$, and $Y=XZ$. Then $X$ and $Y$ are uncorrelated standard normal (in particular log-concave) random variables, but the distribution of $(X,Y)$ is not log-concave.


This is of course also a counterexample to the false theorem one sometimes hears stated that "uncorrelated normal random variables are independent." (The theorem becomes true if "jointly" is inserted in the right place.)


