Metric embeddings, energy integrals and summing operators

DANIEL CARANDO

Buenos Aires

Congreso; Tercera Escuela sobre Análisis Funcional y Geometría; 2016

In this talk we show how some results from the theory of absolutely summing operators can be applied to a problem in metric geometry. We take a compact set $K \subset \mathbb{R}^n$ and endow it 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 new metric space $(K, d_{\alpha})$ may be isometrically embedded on the surface of a Hilbert sphere of radius $r$. Our aim is to estimate this minimun radius, and we do it for several centrally symmetric convex bodies $K$. To this end, we relate this radius with some \emph{energy integrals} on $K$. This allows us to estimate the radius $r$ in terms of the $2 \alpha$-summing norm of certain operators or the mean width of $K$.This is a joint work with Daniel Galicer and Dami\'an Pinasco.