Abstract. We study the convexity and starlikeness of metric balls on Banach spaces when the metric is the quasihyperbolic metric or the distance ratio metric. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.
Abstract. We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain Ω in a Banach space such that there is no geodesic between any given pair of points x, y ∈ Ω. In addition, we prove that if X is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of X, is smooth.
We introduce and study a natural class of variable exponent p spaces, which generalizes the classical spaces p and c0. These spaces will typically not be rearrangementinvariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.
Abstract. We present an equivalent midpoint locally uniformly rotund (MLUR) renorming X of C[0, 1] on which every weakly compact projection P satisfies the equation I − P = 1 + P (I is the identity operator on X). As a consequence we obtain an MLUR space X with the properties D2P, that every non-empty relatively weakly open subset of its unit ball BX has diameter 2, and the LD2P+, that for every slice of BX and every norm 1 element x inside the slice there is another element y inside the slice of distance as close to 2 from x as desired. An example of an MLUR space with the D2P, the LD2P+, and with convex combinations of slices of arbitrary small diameter is also given.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.