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Volume ratios in L p -spaces

Yehoram Gordon, Marius Junge (1999)

Studia Mathematica

There exists an absolute constant c 0 such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that i n f e l l i p s o i d ε B E ( v o l ( B E ) / v o l ( ε ) ) 1 / n c 0 i n f z o n o i d Z B F ( v o l ( B F ) / v o l ( Z ) ) 1 / k . The concept of volume ratio with respect to p -spaces is used to prove the following distance estimate for 2 q p < : s u p F p , d i m F = n i n f G L q , d i m G = n d ( F , G ) c p q n ( q / 2 ) ( 1 / q - 1 / p ) .

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