Geometric and probabilistic analysis of convex bodies with unconditional structures, and associated spaces of operators (Q963679)

The paper deals with geometric properties of unconditional and symmetric norms on \(\mathbb{R}^n\). It is proved that if \(\|\cdot\|\) is unconditional (with respect to the canonical basis \((e_i)_i\)) and \(\|e_i\|=1\) for every \(i\) then for every \(x\) one has \(2\|x\|_2 \leq \|x\| + \|x\|_* \leq \|x\|_1 + \|x\|_{\infty}\), where \(\|x\|_p\) denotes the \(\ell _p\)-norm. If \(\|\cdot\|\) is in addition symmetric then for every \(x\), \(y\) one has \(\|x\| \|y\|_* \leq \max\{\|x\|_1 \|y\|_{\infty},\;\|x\|_{\infty} \|y\|_1\}\). The authors provide many examples and some extensions to the case of two symmetric norms \(\|\cdot\|\) and \(|||\cdot|||\) satisfying \(\|\cdot\|\leq |||\cdot|||\leq \|\cdot\|_*\). They also show applications to the random embeddings of Euclidean spaces into spaces of nuclear operators and to the behavior of the so-called \(\ell\)-norms of operators.