On averaged self-distances in finite dimensional Banach spaces

Abstract

Assume that $\mathfrak{A}$ is a real Banach space of finite dimension $n\ge 2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak{A}$. We show that \[Δ(ν)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,ν(x)\,ν(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb{N}\setminus \{0,1\}\to (0,1]$ 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 `$1$'.

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