LUKK Property And Its Applications

We introduced two new geometric properties by localizing two important geometric properties. Meanwhile, two geometric coefficients were localized, then two new geometric coefficients were introduced to wide the way of describing the geometric structure of Banach space in this thesis. According to the requirement of various theories and application, we introduced a new dentability and the relationship between some denting points and exposed points was also discussed. At last, exposed point was further investigated in Orlicz space. The contents of the paper are organized as follow:First, we respectively localized two important geometric properties UKK , UKK c then introduced two new geometric properties LUKK and LUKK c. Meanwhile, we respectively localized P_X (ε)that is the coefficient of UKK , andΔ_X (ε)that is the modulus of noncompact convexity then introduced P_X^L(ε)that is the modulus of LUKK andΔ^L_X (ε)that is the modulus of LNUC . Next, we studied the implication between LUKK and ( H ), UKK,LNUC . The relationship that P_X^L(ε)andΔ^L_X (ε)satisfy was also discussed, the range of P_X^L(ε)was obtained in l P. At last, we proved that LUKK cwas equivalent toδ₂ condition in Orlicz sequence space.Second, we introduced the concept of weakly locally uniformly denting points and k-w*denting points. Meanwhile, we respectively constructed the relationship between weak dentability, weakly locally uniformly dentability and some classes of smoothness. Giving the criteria for every point on the sphere is a weakly locally uniformly denting point in reflexive Banach space. Furthermore, we studied the relationship between k-w denting points and k-wstrongly exposed points, between k-w*denting points and k-w*strongly exposed points, respectively.Finally, the criteria for every point on the sphere is an exposed point in Banach space, we did further research in exposed point and discussed the relationship between exposed points, extreme points and supporting functionals next.