On averaged self-distances in finite dimensional Banach spaces

Abstract

Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,\nu(x)\,\nu(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb N\setminus \{0,1\}\rightarrow (0,1]$ is a concrete universal function such that $f(n)\sim \frac{2}{\mathrm e n^2\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'.