Tight embedding of subspaces of L p in l n p for even p ∗ Gideon Schechtman

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of lp , p an even integer, needed to almost isometrically contain all k-dimensional subspaces of Lp. In a recent paper [BSS] Batson, Spielman and Srivastava introduced a new method for sparsification of graphs which already proved to have functional analytic applications. Here we bring one more such application. Improving over a result of [BLM] (or see [JS] for a survey on this and related results), we show that for even p and for k of order n any k dimensional subspace of Lp nicely embeds into l n p . This removes a log factor from the previously known estimate. The result in Theorem 2 is actually sharper than stated here and gives the best possible result in several respects, in particular in the dependence of k on n. The theorem of [BSS] we shall use is not specifically stated in [BSS], but is stated as Theorem 1.6 in Srivastava’s thesis [Sr]: Theorem 1 [BSS] Suppose 0 < ε < 1 and A = ∑m i=1 viv T i are given, with vi column vectors in R. Then there are nonnegative weights {si}i=1, at most ⌈k/ε⌉ of which are nonzero, such that, putting à = ∑m i=1 siviv T i , (1− ε)−2xTAx ≤ x Ãx ≤ (1 + ε)xAx for all x ∈ R. ∗AMS subject classification: 46B07 †Supported by the Israel Science Foundation and by the U.S.-Israel Binational Science Foundation