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Extremal properties of half-spaces for log-concave distributions. (English) Zbl 0859.60048
Let $$\mu$$ be a log-concave probability measure on the real line $$\mathbb{R}$$. The author considers the value $$r^n_h(p)=\inf\mu^n(A+hD_n)$$, where $$\mu^n=\mu\times\cdots\times\mu$$ is the product measure in $$\mathbb{R}^n$$, $$D_n=[-1,1]^n$$ is the $$n$$-dimensional cube in $$\mathbb{R}^n$$ and the infimum is over all Borel-measurable sets $$A\subset\mathbb{R}^n$$ of measure $$\mu^n(A)\geq p$$; $$0<p<1$$, $$h>0$$. He gives necessary and sufficient conditions under which this infimum is attained at the half-spaces $$\{x\in\mathbb{R}^n: x_1\leq c\}$$ of measure $$p$$, for all $$p$$ and $$h$$.

##### MSC:
 60G70 Extreme value theory; extremal stochastic processes
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##### References:
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