On averaged self-distances in finite dimensional Banach spaces

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 $$\mathrm{\Delta }(\nu )={\int }_{x\in K}{\int }_{y\in K}\mathrm{\mid }x-y{\mathrm{\mid }}_{\mathfrak{A}}\nu (x)\nu (y)\le 2(1-2^{-n}f(n)),$$ 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 $1$’.