694 research outputs found
A parametric smooth variational principle and support properties of convex sets and functions
No full textWe show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets
Polyhedral direct sums of Banach spaces, and generalized centers of finite sets
No full textA Banach space is said to satisfy (GC) if
the set of minimizers
of the function is nonempty
for each integer , each and each
continuous nondecreasing coercive real-valued function on .
We study stability of certain polyhedrality properties under making direct sums,
in order to be able to use
results from a paper by Fonf, Lindenstrauss and the author to
show that if satisfies (GC) and an appropriate polyhedrality property
then the function space satisfies (GC) for every topological space
. This generalizes the author's result from 1997, proved for
finite dimensional polyhedral spaces . Moreover, under more restrictive conditions
on and , the mappings on ()
are continuous in the Hausdorff metric for each compact
On differentiability of convex operators
No full textThe main known results on differentiability of continuous convex operators ff from a Banach space XX to an ordered Banach space YY are due to J.M. Borwein and N.K. Kirov. Our aim is to prove some “supergeneric” results, i.e., to show that, sometimes, the set of Gâteaux or Fréchet nondifferentiability points is not only a first-category set, but also smaller in a stronger sense. For example, we prove that if YY is countably Daniell and the space L(X,Y)L(X,Y) of bounded linear operators is separable, then each continuous convex operator f:X→Yf:X→Y is Fréchet differentiable except for a Γ-null angle-small set. Some applications of such supergeneric results are shown
On connections between delta-convex mappings and convex operators
No full textWe study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping F:(a, b) -> Y is the difference of two continuous convex operators whenever Y belongs to a large class of Banach lattices which includes all L-p (mu) spaces (1 <= p <= infinity). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented
On vector functions of bounded convexity
No full textsummary:Let be a normed linear space. We investigate properties of vector functions of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity is equal to the variation of on . As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping
Quasi uniform convexity : Revisited
No full textQuasi uniform convexity (QUC) is a geometric property of Banach spaces, introduced in 1973 by J.R. Calder et al., which implies existence of Chebyshev centers for bounded sets. We extend and strengthen some known results about this property. We show that (QUC) is equivalent to existence and continuous dependence (in the Hausdorff metric) of Chebyshev centers of bounded sets. If X is (QUC) then the space C(K;X) of continuous X-valued functions on a compact K is (QUC) as well. We also show that a sufficient condition introduced by L. Pevac already implies (QUC), and we provide a couple of new sufficient conditions for (QUC). Together with Chebyshev centers, we consider also asymptotic centers for bounded sequences or nets (of points or sets)
One-point location problems from a very general point of view
No full textGiven a nonempty set in a metric space , we consider the
problem of minimizing a function of the form
where is a monotone functional on the set \lip(A) of all nonnegative
-Lipschitz functions on , and is the
(-Lipschitz) function .
The minimizers (if any) of
over are called
{\em -centers} of the set .
\par
We present an existence theorem for -centers, based on compactness in
the so called -topology and on the Fatou property of .
We discuss sufficient conditions for the assumptions being
satisfied. The last section is devoted to significant particular cases:
generalized integral medians, generalized
Chebyshev centers, and ``generalized centers with neglect''''
Spaces of d.c. mappings on arbitrary intervals
No full textLet X be a Banach space. Using derivatives in the sense of vector distributions, we show that the space DC([0, 1], X) of all d.c. mappings from [0, 1] into X, in a natural norm, is isomorphic to the space M-bv([0, 1], X) of all vector measures with bounded variation. The same is proved for the space BDCb((0,infinity), X) of all bounded d.c. mappings with a bounded control function. The result for the space DC([0, 1], R) of all continuous d.c. functions was (essentially) proved by M.Zippin (2000) by a quite different method. The space BDCb((0,infinity), R) consists of all differences of two bounded convex functions. Internal characterizations of its members were given by O. Bohme (1985), but our characterization of its Banach structure is new
On compositions of d.c. functions and mappings
No full textA d.c. (delta-convex) function on a normed linear space is a function representable as a
difference of two continuous convex functions.
We show that an infinite dimensional analogue of Hartman''s theorem on stability of d.c. functions under compositions does not hold in general.
However, we prove that it holds in some
interesting particular cases.
Our main results about compositions are proved in the more general context of d.c.
mappings between normed linear spaces
On extensions of d.c. functions and convex functions
No full textWe show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J. Borwein, V. Montesinos and J. Vanderwerff
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