Energy integrals and metric embedding theory

Abstract

For some centrally symmetric convex bodies $K\subset {\mathbb{R}}^n$, we study the energy integral $$\sup {\int }_K{\int }_K\Vert x-y{\Vert }_r^{p}d\mu (x)d\mu (y),$$ 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 $1<q\le 2$) 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 $R$.

17 pages, Accepted in International Mathematics Research Notices