The classical Dvoretzky–Rogers lemma provides a deterministic
algorithm by which, from any set of isotropic vectors in Euclidean
-space, one can
select a subset of
vectors whose determinant is not too small. Pełczyński 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
independent
random vectors in
,
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 Pełczyński and Szarek, and provide a
lower bound for the probability that the volume of such a random parallelotope is
large.
Keywords
isotropic vectors, John's theorem, Dvoretzky–Rogers lemma,
decomposition of the identity, volume