Some estimates for the normal structure coefficient in Banach spaces (Q1814412)

If \(C\) is a non-empty bounded subset of a Banach space \(X\), let \[ r(C)=\inf_{y\in C} \sup_{x\in C}\| x-y\| \] and \(N(X)=\inf\{\text{diam} C\}\), the inf taken over all closed convex sets \(C\) having \(r(C)=1\). The space \(X\) is said to have uniform normal structure if \(N(X)>1\). In this paper estimates of \(N(X)\) related to the modulus of convexity and modulus of smoothness of \(X\) are obtained which, in particular, imply several known results concerning uniform normal structure of certain Banach spaces.