On k-WUR and its generalizations
Abstract
We introduce two notions called k-weakly uniform rotundity (k-WUR) and k-weakly locally uniform rotundity (k-WLUR) in real Banach spaces. These are natural generalizations of the well-known concepts k-UR and WUR. By introducing two best approximation notions namely k-weakly strong Chebyshevity and k-weakly uniform strong Chebyshevity, we generalize some of the existing results to k-WUR and k-WLUR spaces. In particular, we present characterizations of k-WUR spaces in terms of k-weakly uniformly strong Chebyshevness. Also, the inheritance of the notions k-WUR and k-WLUR by quotient spaces are discussed. Further, we provide a necessary and sufficient condition for an infinite p-product space to be k-WUR (respectively, k-WLUR). As a consequence, we observe that the notions WUR and k-WUR coincide for an infinite p-product of a Banach space.
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