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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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