Metric geometry and energy integrals for convex bodies

CARANDO DANIEL; DAMIÁN PINASCO; DANIEL GALICER

Valencia

Workshop; Workshop on Functional Analysis Valencia 2013 on the occasion of the 60th birthday of Andreas Defant; 2013

Let $K \subset \mathbb{R}^n$ be a compact set endowed with the metric $d_{\alpha}(x,y)=|x - y|^{\alpha}$, where $0 < \alpha < 1$. A classical result of Schoenberg and von Neumann asserts that there exist a minimum $r$ for which the metric space $(K, d_{\alpha})$ may be isometrically embedded on the surface of a Hilbert sphere of radius $r$. We provide estimates of these radii for several centrally symmetric convex bodies $K$. To this end, we study the energy integral $$\sup \int_{K} \int_{K} |x - y|^{2 \alpha} d\mu(x) d\mu(y),$$ where the supremum runs over all finite signed Borel measures $\mu$ on $K$ of total mass one. We bound this value by the mean width of $K$ or the $2 \alpha$-summing norm of certain operator. In the case where $K$ is an ellipsoid or $K = B_q^n$, the unit ball of $\ell_q^n$ (for $1 \leq q \leq 2$), we obtain the correct asymptotical behavior of the least possible radius. Joint work with Daniel Carando and Dami\'an Pinasco.