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Moduli of rotundity and smoothness for convex bodies. (English) Zbl 0804.52001

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.

MSC:

52A05 Convex sets without dimension restrictions (aspects of convex geometry)
46B20 Geometry and structure of normed linear spaces
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