@article{6234aff022ca4257b62c4f110e33c71b,
title = "Sharp inequalities for the mean distance of random points in convex bodies",
abstract = "For a convex body K⊂Rd the mean distance Δ(K)=E|X1−X2| is the expected Euclidean distance of two independent and uniformly distributed random points X1,X2∈K. Optimal lower and upper bounds for ratio between Δ(K) and the first intrinsic volume V1(K) of K (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.",
keywords = "Geometric extremum problem, Geometric inequalities, Integral geometry, Mean distance, Mean width, Riesz's rearrangement inequality",
author = "Gilles Bonnet and Anna Gusakova and Christoph Th{\"a}le and Dmitry Zaporozhets",
note = "Funding Information: The work of GB and AG was partially supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-Dimensional Phenomena in Probability-Fluctuations and Discontinuity. The work of DZ was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” and by RFBR and DFG according to the research project 20-51-12004 . Funding Information: This project has been initiated when DZ was visiting Ruhr University Bochum in September and October 2019. Financial support of the German Research Foundation ( DFG ) via Research Training Group RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity is gratefully acknowledged. We also thank an anonymous referee for insightful comments and remarks which helped us to further improve our paper. We also thank Uwe B{\"a}sel for pointing us to the correct value for in (5) . Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",
year = "2021",
month = aug,
day = "6",
doi = "10.1016/j.aim.2021.107813",
language = "English",
volume = "386",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Elsevier",
}