Zanco, Clemente; Zucchi, Adele Moduli of rotundity and smoothness for convex bodies. (English) Zbl 0804.52001 Boll. Unione Mat. Ital., VII. Ser., B 7, No. 4, 833-855 (1993). Let \((X,\| \cdot \|)\) be a normed real linear space of dimension at least 2. A rooted body in \(X\) is an ordered pair \((K,r)\) such that \(K\) is a bounded, closed, convex subset of \(X\) and \(r\) is an interior point of \(K\). The \(M\)-diameter of \((K,r)\) is defined by \[ d_ M(K,r) = \sup \{q_{K-r} (x - y):\;x,y \in K\}, \] with \(q_{K-r}\) denoting the Minkowski functional of \(K - r\). The authors prove that, for \(1 \leq p < + \infty\), there exists in \(\ell^ p\) a rooted body whose \(M\)-diameter is not attained. Based on the notion of the \(M\)-diameter, they define the modulus of rotundity \(\delta_{(K,r)}\) and the modulus of smoothness \(\rho_{(K,r)}\) of \((K,r)\) and provide characterizations of the rotundity and the uniform smoothness of \(K\) in terms of the families of functions \(\{\delta_{(K,r)}\}_{r \in \text{int} K}\) and \(\{\rho_{(K,r)}\}_{r \in \text{int} K'}\) respectively. Reviewer: J.-E.MartĂnez-Legaz (Barcelona) Cited in 2 ReviewsCited in 5 Documents MSC: 52A05 Convex sets without dimension restrictions (aspects of convex geometry) 46B20 Geometry and structure of normed linear spaces Keywords:convex body; modulus of rotundity; modulus of smoothness PDFBibTeX XMLCite \textit{C. Zanco} and \textit{A. Zucchi}, Boll. Unione Mat. Ital., VII. Ser., B 7, No. 4, 833--855 (1993; Zbl 0804.52001)