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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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