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Special orthogonal splittings of $L_1^{2k}$

Gideon Schechtman
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Abstract

We show that for each positive integer $k$ there is a $k\times k$ matrix $B$ with $\pm 1$ entries such that putting $E$ to be the span of the rows of the $k\times 2k$ matrix $[\sqrt{k}I_k,B]$, then $E,E^{\bot}$ is a Kashin splitting: The $L_1^{2k}$ and the $L_2^{2k}$ are universally equivalent on both $E$ and $E^{\bot}$. Moreover, the probability that a random $\pm 1$ matrix satisfies the above is exponentially close to 1.

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Gideon Schechtman. 2003-03-09. Special orthogonal splittings of $L_1^{2k}$. https://arxiv.org/abs/math/0301275

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