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Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural conditions that ensure that the integral $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ is finite. If $\pi$ is radially decreasing, this is equivalent to the condition $\mathbb{E}\big[ \|X\|^{d} \big] < \infty$. Are there smooth densities verifying this moment condition such that $\iint_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv = \infty$ ?

Update: a solution can now be found on mathoverflow.

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