An upper bound q(c) for the best, under equivalent renorming, possible power type of the modulus of smoothness of a Banach space with modulus of convexity satisfying δX(ε) cε2, is found. The estimate is asymptotically sharp and is expressed in terms of linear fractional function q(c). © 2006 Elsevier Inc. All rights reserved.

We present some sufficient conditions for which a Banach space X has normal structure in terms of the modulus of U-convexity, modulus of W∗-convexity, and the coefficient R 1, X , which generalized some well-known results. Furthermore the relationship between modulus of convexity, modulus of smoothness, and Gao’s constant is considered, meanwhile the exact value of Milman modulus has been obtai...

In optimization, the notion of a partly smooth objective function is powerful for applications in algorithmic convergence and postoptimality analysis, yet complex to define. A shift focus first-order optimality conditions reduces concept simple constant-rank condition. this view, partial smoothness extends more general variational systems, encompassing particular saddlepoint operators underlyin...

In Hilbert space L2(Rn), we prove the equivalence between the mod-ulus of smoothness and the K-functionals constructed by the Sobolev space cor-responding to the Fourier transform. For this purpose, Using a spherical meanoperator.