TY - JOUR

T1 - Sharp inequalities for the mean distance of random points in convex bodies

AU - Bonnet, Gilles

AU - Gusakova, Anna

AU - Thäle, Christoph

AU - Zaporozhets, Dmitry

PY - 2021/8/6

Y1 - 2021/8/6

N2 - 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.

AB - 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.

KW - Geometric extremum problem

KW - Geometric inequalities

KW - Integral geometry

KW - Mean distance

KW - Mean width

KW - Riesz's rearrangement inequality

UR - http://www.scopus.com/inward/record.url?scp=85108219503&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107813

DO - 10.1016/j.aim.2021.107813

M3 - Article

AN - SCOPUS:85108219503

VL - 386

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -