关于Dvoretzky-Rogers引理的体积界
On the volume bound in the Dvoretzky--Rogers lemma
摘要：
经典的Dvoretzky-Rogers引理提供了一种确定性算法，通过该算法，从欧几里得$d$ -空间中的任何迷向向量集合中，可以选择行列式不太小的$d$向量的子集。随后，Pelczyovski和Szarek改进了这个下界的一个因素取决于维度和数量的向量。另一方面，Mesquivarov确定了由$\mathbb{R}^d$中的$d$独立随机向量所跨越的平行体的体积的平方的期望，每个向量都是根据各向同性度量来选择的。我们扩展的Pastivarov的结果，一类更一般的概率测度，这产生的体积限制在Dvoretzky-罗杰斯引理，事实上，等于期望的平方体积随机平行空间跨越的迷向向量。这使我们能够给出一个概率的证明Pelczyovski和Szarek的改进，并提供了一个下界的概率，这种随机平行体的体积是大的。
英文摘要：
The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by which, from any set of isotropic vectors in Euclidean $d$-space, one can select a subset of $d$ vectors whose determinant is not too small. Subsequently, Pelczyński and Szarek improved this lower bound by a factor depending on the dimension and the number of vectors. Pivovarov, on the other hand, determined the expectation of the square of the volume of parallelotopes spanned by $d$ independent random vectors in $\mathbb{R}^d$, each one chosen according to an isotropic measure. We extend Pivovarov's result to a class of more general probability measures, which yields that the volume bound in the Dvoretzky--Rogers lemma is, in fact, equal to the expectation of the squared volume of random parallelotopes spanned by isotropic vectors. This allows us to give a probabilistic proof of the improvement of Pelczyński and Szarek, and provide a lower bound for the probability that the volume of such a random parallelotope is large.
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