CONICET | Buscador de Institutos y Recursos Humanos

For some centrally symmetric convex bodies $Ksubset mathbb R^n$, we study the energy integral$$sup int_{K} int_{K} |x - y|_r^{p}, dmu(x) dmu(y),$$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~$R$.