On the concentration of measure and the \(L^1\)-norm (Q741095)

Introducing a certain quantity \(\alpha ^* (M)\), where \(M\subset L^1 =: L^1 (\Xi ,\mu) \) with \( (\Xi ,\mu) \) as some measurable space. \textit{Y. Benyamini} et al. [Constr. Approx. 36, No. 3, 399--431 (2012; Zbl 1262.41017), p. 246] proposed a conjecture on the maximal value of \(\alpha^*\) over all infinite-dimensional subspaces of \(L^1\). In the present paper the authors decide this conjecture. The proof presented here is based on the concentration of measure phenomenon (see e.g. [\textit{M. Ledoux}, The concentration of measure phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0995.60002)]).