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Generalized volume ratios and the Banach-Mazur distance. (English. Russian original) Zbl 1048.46019
Math. Notes 70, No. 6, 838-846 (2001); translation from Mat. Zametki 70, No. 6, 918-926 (2001).
The article is devoted to studying the classical and modified Banach-Mazur distances. Let \(X\) and \(Y\) be \(n\)-dimensional normed spaces. Let \[ Vr(X,Y) = \left(\inf\left\{\frac{\text{vol}B_X}{\text{vol}UB_Y}\,:\, UB_Y\subset B_X\right\}\right)^{1/n}, \] where \(B_X\) and \(B_Y\) are the unit balls in the spaces \(X\) and \(Y\). For \(Y=l_n^2\), the volume ratio \(Vr(X,l_n^2)\) coincides with the classical volume ratio. The main aim is to establish a relation between \(\partial(X,Y)\) and \(Vr(X,Y)\), calculate the volume ratios for the spaces \(l_n^p\) and their sums, determine \(\sup\partial(X,l_n^p)\) and estimate \(\sup Vr(X,Y)\). Here \(\partial(X,Y)\) denotes the modified Banach-Mazur distance.

MSC:
46B07 Local theory of Banach spaces
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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