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Sections of the unit ball of \(\ell ^ n_ p\). (English) Zbl 0667.46004

Let \(B^ n_ p=\{(x_ i)\in {\mathbb{R}}^ n|\) \(\sum^{n}_{i=1}| x_ i|^ p\leq 1\}\), \(1\leq p\leq +\infty\). The main result of the paper (see Theorem and Theorem II.2) says that \[ h: [1,+\infty]\to {\mathbb{R}}_+,\quad h(p):=vol(E^ k\cap B^ n_ p)/vol(B^ k_ p) \] is increasing. In particular, \(vol(E^ k\cap B^ n_ p)\leq vol(B^ k_ p)\) for \(1\leq p\leq 2\) and \(vol(E^ k\cap B^ n_ p)\geq vol(B^ k_ p)\) for \(2\leq p\leq +\infty\). These results extend the result of J. D. Vaaler [Pac. J. Math. 83, 543-553 (1979; Zbl 0465.52011)] that the volume of sections of the cube \([-,]^ n\) by k-dimensional subspaces of \({\mathbb{R}}^ n\) is always bigger than 1. The authors give some interesting applications to Number Theory (Theorems III.1 and III.2) and to volume ratio numbers (Theorem III.3) and they derive lower estimates for the volume of sections (Theorems III.5 and III.8).
Reviewer: J.Boos

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces

Citations:

Zbl 0465.52011
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References:

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