More on embedding subspaces of \(L_ p\) in \(l^ n_ r\). (English) Zbl 0659.46021

The author carries further his own work and those of others on embeddings of finite dimensional subspaces of \(L_ p(0,1)\) into \(\ell^ n_ r\).
In the present paper the author takes K-embeddings: Given two normed spaces X and Y and \(1\leq K<\infty\), X is said to K-embed into Y (denoted \(X\hookrightarrow^{K}Y)\) if there is a one-one linear operator \[ T:X\to T(X)\subseteq Y\quad with\quad \| T\| \| T^{-1}\| \leq K. \] Taking X as an m-dimensional subspace of \(L_ p\) and Y as \(\ell^ n_ r\), K-embeddings of X into Y are studied. The main concern is to find as to how small n can be (K,r,p,m given). Various bounds are achieved. The results obtained are improvements of earlier results. Some open problems are stated and analysed at the end.
The author also acknowledges (private communication) the improvement of the results of the current paper by Bourgain, Lindenstrauss and Milman.
Reviewer: Shaligram Singh


46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
Full Text: Numdam EuDML


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