Energy Integrals and Metric Embedding Theory

Abstract

For some centrally symmetric convex bodies K ⊂ Rn, we study the energy integral sup K K x − yp r 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 = Bn q , the unit ball of n q (for 1 < q ≤ 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 Rn the Euclidean distance d2. For 0 <α< 1, we estimate the minimum R for which the snowflaked metric space (K,dα 2 ) may be isometrically embedded on the surface of a Hilbert sphere of radius R. 1 I