Sufficient conditions for uniform normal structure of Banach spaces and their duals (Q874967)

In general, having uniform normal structure is not a self-dual property of Banach spaces [\textit{M.\,A.\thinspace Smith} and \textit{B. Turett}, J.~Aust. Math. Soc., Ser. A 48, No. 2, 223--234 (1990; Zbl 0763.46013)]. However, in the paper under review, the author proves that several conditions known to imply uniform normal structure of a Banach space also imply uniform normal structure of its dual space. In particular, the author considers five different parameters: an arc length parameter, a generalized von Neumann--Jordan constant, a parametrized James constant, the modulus of smoothness, and a modulus of \(U\)-convexity. For an example of a typical theorem, let \(X\) be a Banach space and let \(R_G(X) = \inf\{\ell(S_Y) - r(S_Y): Y\;\text{is a two-dimensional subspace of}\;X\}\) where \(\ell(S_Y)\) is the circumference of the unit ball of \(Y\) and \(r(S_Y)\) is the supremum of the perimeters of the parallelograms inscribed in the unit ball of \(Y\). In [Taiwanese J.~Math. 5, No. 2, 353--366 (2001; Zbl 0984.46011)], \textit{J. Gao} proved that \(R_G(X)>0\) implies that \(X\) has uniform normal structure. The author shows that \(R_G(X)>0\) also implies that \(X^*\) has uniform normal structure.