2,963 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}
Higher order extension of L\"owner's theory: Operator -tone functions
The new notion of operator/matrix -tone functions is introduced, which is
a higher order extension of operator/matrix monotone and convex functions.
Differential properties of matrix -tone functions are shown.
Characterizations, properties, and examples of operator -tone functions are
presented. In particular, integral representations of operator -tone
functions are given, generalizing familiar representations of operator monotone
and convex functions.Comment: final version, 33 page
Local strong maximal monotonicity and full stability for parametric variational systems
The paper introduces and characterizes new notions of Lipschitzian and
H\"olderian full stability of solutions to general parametric variational
systems described via partial subdifferential and normal cone mappings acting
in Hilbert spaces. These notions, postulated certain quantitative properties of
single-valued localizations of solution maps, are closely related to local
strong maximal monotonicity of associated set-valued mappings. Based on
advanced tools of variational analysis and generalized differentiation, we
derive verifiable characterizations of the local strong maximal monotonicity
and full stability notions under consideration via some positive-definiteness
conditions involving second-order constructions of variational analysis. The
general results obtained are specified for important classes of variational
inequalities and variational conditions in both finite and infinite dimensions
Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers
In this work, using Moreau envelopes, we define a complete metric for the set
of proper lower semicontinuous convex functions. Under this metric, the
convergence of each sequence of convex functions is epi-convergence. We show
that the set of strongly convex functions is dense but it is only of the first
category. On the other hand, it is shown that the set of convex functions with
strong minima is of the second category
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