Energy integrals and metric embedding theory

For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $\sup \int_{K} \int_{K} \|x - y\|_r^{p}\, dμ(x) dμ(y),$ where the supremum runs over all finite signed Borel measures $μ$ 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 $0 < α< 1$, we estimate the minimum $R$ for which the snowflaked metric space $(K, d_2^α)$ may be isometrically embedded on the surface of a Hilbert sphere of radius $R$.

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