Energy integrals and metric embedding theory

Abstract

For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $$ \sup \int_{K} \int_{K} \|x - y\|_r^{p}\, d\mu(x) d\mu(y), $$ where the supremum runs over all finite signed Borel measures $\mu$ on $K$ of total mass one. In the case where $K = B_q^n$, the unit ball of $\ell_q^n$ (for $1 < q \leq 2$) or an ellipsoid, we obtain the exact value or the correct asymptotical behavior of the supremum of these integrals. We apply these results to a classical embedding problem in metric geometry. We consider in $\mathbb R^n$ the Euclidean distance $d_2$. For $0 < \alpha < 1$, we estimate the minimum $R$ for which the snowflaked metric space $(K, d_2^{\alpha})$ may be isometrically embedded on the surface of a Hilbert sphere of radius $R$.