Online embedding of metrics

We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces, a tight exponential upper bound on embedding into the line, and a $(1+\epsilon )$-distortion embedding in ${\mathrm{\ell }}_{\mathrm{\infty }}$ of a suitably high dimension.