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The theorems of Caratheodory and Gluskin for $0<p<1$

Jesus Bastero·J. Buernes·A. Pena
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Abstract

In this note we investigate some aspects of the local structure of finite dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional $p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show that such bound is optimum.

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Jesus Bastero, J. Buernes, A. Pena. 1992-09-10. The theorems of Caratheodory and Gluskin for $0<p<1$. https://arxiv.org/abs/math/9209213

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