  A generalization of Lozanovskii's result is proved. Let E be $k$-dimensional
subspace of an $n$-dimensional Banach space with unconditional basis. Then
there exist $x_1,..,x_k \subset E$ such that $B_E \p \subset \p
absconv\{x_1,..,x_k\}$ and \[ \kla \frac{{\rm vol}(absconv\{x_1,..,x_k\})}{{\rm
vol}(B_E)} \mer^{\frac{1}{k}} \kl \kla e\p \frac{n}{k} \mer^2 \pl .\] This
answers a question of V. Milman which appeared during a GAFA seminar talk about
the hyperplane problem. We add logarithmical estimates concerning the
hyperplane conjecture for proportional subspaces and quotients of Banach spaces
with unconditional basis.
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