Geometry of spaces between polytopes and related zonotopes
Let \(k\leq N\) be positive integers, \((a_i)_{i\leq\mathbb{N}}\subset \mathbb{R}^n\) be a sequence of vectors spanning \(\mathbb{R}^n\), and let \(\|x\|_{k,q}= (\sum_{1\leq i\leq k}|\langle x,a_i\rangle|^{*q})^{{1\over q}}\), where \((|\langle x,a_i\rangle|^*)_i\) is the nonincreasing rearrangement of the sequence \((|\langle x,a_i\rangle|)_i\). The normed space \((\mathbb{R}^n,\|\cdot\|_{k,q})\) is denoted by \(X_{k,q}\), its unit ball is denoted by \(B_{k,q}\). The authors investigate the geometry of these spaces and their duals. These spaces are of interest because they generalize the class of dual spaces of zonotopes in a natural way. For \(k= N\) and \(q= 1\), the unit balls of \(X^*_{k,q}\) are zonotopes, and for \(k= 1\) and \(q= 1\), the spaces \(X_{k,q}\) range over all possible spaces with a polytopal unit ball with no more than \(2N\) facets. The authors provide upper and lower estimates for the volumes of \(B_{k,q}\) and their lower-dimensional sections. They determine the dimension of almost Euclidean sections obtaining a Dvoretzky-type theorem. Dealing with random parameters of the spaces, the authors apply the probabilistic results obtained by them in [Ann. Probab. 30, 1833-1853 (2002; Zbl 1016.60008)]. In the last section, the authors consider the important particular case when \(N= n\), \((a_i)_i= (e_i)_i\) is the canonical basis of \(\mathbb{R}^n\), and \(q= 1\). In this case, the space \(X_{k,1}\) becomes an interpolation space between \(\ell_1\) and \(\ell_\infty\). The authors obtain asymptotically sharp estimates for the most important geometric parameters of \(X_{k,1}\) such as type and cotype constants, projection constant, volume ratio, \(p\)-summing norms, etc.
scientific article; zbMATH DE number 1888298
scientific article