3,521 research outputs found
New Moduli for Banach Spaces
Modifying the moduli of supporting convexity and supporting smoothness, we
introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti
of right-angled triangles (defined via so-called quasi-orthogonality). These
triangles have two boundary points of the unit ball of a Banach space as
endpoints of their hypotenuse, and their third vertex lies in a supporting
hyperplane of one of the two other vertices. Among other things it is our goal
to quantify via such triangles the local deviation of the unit sphere from its
supporting hyperplanes. We prove respective Day-Nordlander type results,
involving generalizations of the modulus of convexity and the modulus of
Bana\'{s}
More on convexity and smoothness of operators
AbstractLet X and Y be Banach spaces and T:YâX be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus ÎŽT of convexity and the T-modulus ÏT of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a FigielâPisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators
Markov convexity and local rigidity of distorted metrics
It is shown that a Banach space admits an equivalent norm whose modulus of
uniform convexity has power-type p if and only if it is Markov p-convex.
Counterexamples are constructed to natural questions related to isomorphic
uniform convexity of metric spaces, showing in particular that tree metrics
fail to have the dichotomy property.Comment: 47 pages, full version, replacing the previous version which was an
announcemen
On proximal mappings with Young functions in uniformly convex Banach spaces
It is well known in convex analysis that proximal mappings on Hilbert spaces
are -Lipschitz. In the present paper we show that proximal mappings on
uniformly convex Banach spaces are uniformly continuous on bounded sets.
Moreover, we introduce a new general proximal mapping whose regularization term
is given as a composition of a Young function and the norm, and formulate our
results at this level of generality. It is our aim to obtain the corresponding
modulus of uniform continuity explicitly in terms of a modulus of uniform
convexity of the norm and of moduli witnessing properties of the Young
function. We also derive several quantitative results on uniform convexity,
which may be of interest on their own.Comment: Accepted in J. Convex Ana
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