zbMATH — the first resource for mathematics

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 \(h\).

60G70 Extreme value theory; extremal stochastic processes
Full Text: DOI
[1] BOBKOV, S. G. 1993. Isoperimetric problem for uniform enlargement. Technical Report 394, Dept. Statistics, Univ. North Carolina. · Zbl 0873.60003
[2] BOBKOV, S. G. 1994. Isoperimetric inequalities for distributions of exponential ty pe. Ann. Probab. 22 978 994. · Zbl 0808.60023 · doi:10.1214/aop/1176988737
[3] BORELL, C. 1974. Convex measures on locally convex spaces. Ark. Mat. 12 239 252. · Zbl 0297.60004 · doi:10.1007/BF02384761
[4] BORELL, C. 1975. The Brunn Minkowski inequality in Gauss space. Invent. Math. 30 207 216. · Zbl 0311.60007 · doi:10.1007/BF01425510
[5] KURATOWSKI, A. 1966. Topology 1. Academic Press, New York. · Zbl 0158.40901
[6] SUDAKOV, V. N. and TSIREL’SON, B. S. 1978. Extremal properties of half-spaces for spheri cally invariant measures. J. Soviet Math. 9 9 18. Translated from Zap. Nauchn. Z. Z. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 41 1974 14 24 in Russian. · Zbl 0395.28007 · doi:10.1007/BF01086099
[7] TALAGRAND, M. 1991. A new isoperimetric inequality and the concentration of measure Z. phenomenon. Israel Seminar GAFA. Lecture Notes in Math. 1469 94 124. Springer, Berlin. · Zbl 0818.46047 · doi:10.1007/BFb0089217
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.