Approximation of zonoids by zonotopes
Let X be an n-dimensional normed space and \(Z\subset X\). Z is called a zonotop if there are segments \((=\) one dimensional compact convex subsets) \(I_ 1,...,I_ N\) such that \(Z=I_ 1+...+I_ N\). A set \(B\subset X\) is a zonoid if for every \(\epsilon >0\) there is a zonotop Z such that \(B\subset Z\subset (1+\epsilon)B.\) In this case the number N(B,\(\epsilon)\) is defined to be the smallest N such that a zonotop Z with \(B\subset Z\subset (1+\epsilon)B\) can be found which is the sum of N segments. It has been shown by \textit{T. Figiel}, \textit{J. Lindenstrauß} and \textit{V. D. Milman} [Acta math. 139, 53-94 (1977; Zbl 0375.52002)] that \[ N(B^ n,\epsilon)\leq cn\epsilon^{-2} \log (1/\epsilon), \] where \(B^ n\) is the n-dimensional Euclidean ball. \textit{Y. Gordon} proved [Isr. J. Math. 50, No.4, 265-289 (1985; Zbl 0663.60034)] that the right hand side of this inequality may be replaced by \(cn\epsilon^{- 2}.\) The main results of the present paper are the following: - \(N(B,\epsilon)\leq cn\epsilon^{-(2+\tau)}\) for every \(\tau >0\) and every B which is the unit ball of a uniformly convex norm in \({\mathbb{R}}^ n\); c depends only on \(\tau\) and the modulus of convexity; - \(N(B,\epsilon)\leq cn\epsilon^{(2+\tau)}(\log n)^ 3\) for arbitrary zonoids B in \({\mathbb{R}}^ n\) and \(\tau >0\), where c depends only on \(\tau\) ; - for fixed n and \(\tau >0\) one has \(N(B^ n,\epsilon)\geq c(n)\epsilon^{-2+r}\) for all \(\epsilon >0\); the reader is also referred to \textit{J. Bourgain} and \textit{J. Lindenstrauß} [Israel J. of Math. 64, No.1, 25-31 (1988; Zbl 0667.52001)], where more information concerning the asymptotic behaviour of \(N(B^ n,\epsilon)\) when \(\epsilon\) \(\to 0\) is obtained. The importance of zonoids B and the numbers N(B,\(\epsilon)\) for Banach space theory stems from the fact that one is let naturally to these notions when considering finite dimensional subspaces X of \(L_ r(0,1)\) and \(\epsilon\)-isomorphisms from X to subspaces of \(\ell^ N_ r\) with ``not too large'' N. The connections of the new results to this problem are indicated. The methods applied are a refinement of those introduction by \textit{G. Schechtman} [Compos. Math. 61, 159-169 (1987; Zbl 0659.46021)]. They also depend on factorization techniques due to Pisier and the empirical distribution method.
scientific article; zbMATH DE number 4117209
scientific article