493,032 research outputs found
Convex-Cyclic Matrices, Convex-Polynomial Interpolation & Invariant Convex Sets
We define a convex-polynomial to be one that is a convex combination of the
monomials . This paper explores the intimate connection
between peaking convex-polynomials, interpolating convex-polynomials, invariant
convex sets, and the dynamics of matrices. In particular, we use these
intertwined relations to both prove which matrices are convex-cyclic while at
the same time proving that we can prescribe the values and a finite number of
the derivatives of a convex-polynomial subject to certain natural constraints.
These properties are also equivalent to determining those matrices whose
invariant closed convex sets are all invariant subspaces. Our characterization
of the convex-cyclic matrices gives a new and correct proof of a similar result
by Rezaei that was stated and proven incorrectly
Convex billiards on convex spheres
In this paper we study the dynamical billiards on a convex 2D sphere. We
investigate some generic properties of the convex billiards on a general convex
sphere. We prove that generically, every periodic point is either
hyperbolic or elliptic with irrational rotation number. Moreover, every
hyperbolic periodic point admits some transverse homoclinic intersections. A
new ingredient in our approach is that we use Herman's result on Diophantine
invariant curves to prove the nonlinear stability of elliptic periodic points
for a dense subset of convex billiards.Comment: 20 pages, 2 figure
Projection and Convolution Operations for Integrally Convex Functions
This paper considers projection and convolution operations for integrally
convex functions, which constitute a fundamental function class in discrete
convex analysis. It is shown that the class of integrally convex functions is
stable under projection, and this is also the case with the subclasses of
integrally convex functions satisfying local or global discrete midpoint
convexity. As is known in the literature, the convolution of two integrally
convex functions may possibly fail to be integrally convex. We show that the
convolution of an integrally convex function with a separable convex function
remains integrally convex. We also point out in terms of examples that the
similar statement is false for integrally convex functions with local or global
discrete midpoint convexity.Comment: 24 page
A Convex Stone-Weierstrass Theorem & Applications
A convex-polynomial is a convex combination of the monomials . This paper establishes that the convex-polynomials on
are dense in and weak dense in , precisely when
. It is shown that the convex-polynomials are dense in
precisely when , where is a compact
subset of the real line. Moreover, the closure of the convex-polynomials on
are shown to be the functions that have a convex-power series
representation.
A continuous linear operator on a locally convex space is
convex-cyclic if there is a vector such that the convex hull of the
orbit of is dense in . The above results characterize which
multiplication operators on various real Banach spaces are convex-cyclic. It is
shown for certain multiplication operators that every closed invariant convex
set is a closed invariant subspace
On Some Generalized Polyhedral Convex Constructions
Generalized polyhedral convex sets, generalized polyhedral convex functions
on locally convex Hausdorff topological vector spaces, and the related
constructions such as sum of sets, sum of functions, directional derivative,
infimal convolution, normal cone, conjugate function, subdifferential, are
studied thoroughly in this paper. Among other things, we show how a generalized
polyhedral convex set can be characterized via the finiteness of the number of
its faces. In addition, it is proved that the infimal convolution of a
generalized polyhedral convex function and a polyhedral convex function is a
polyhedral convex function. The obtained results can be applied to scalar
optimization problems described by generalized polyhedral convex sets and
generalized polyhedral convex functions
On Basic Operations Related to Network Induction of Discrete Convex Functions
Discrete convex functions are used in many areas, including operations
research, discrete-event systems, game theory, and economics. The objective of
this paper is to investigate basic operations such as direct sum, splitting,
and aggregation that are related to network induction of discrete convex
functions as well as discrete convex sets. Various kinds of discrete convex
functions in discrete convex analysis are considered such as integrally convex
functions, L-convex functions, M-convex functions, multimodular functions, and
discrete midpoint convex functions.Comment: 42 pages. arXiv admin note: text overlap with arXiv:1907.0916
On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets
The Krein-Milman theorem (1940) states that every convex compact subset of a
Hausdorfflocally convex topological space, is the closed convex hull of its
extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the
general framework of -convexity. Under general conditions on the class of
functions , the Krein-Milman-Ky Fan theorem asserts then, that every
compact -convex subset of a Hausdorff space, is the -convex hull of
its -extremal points. We prove in this paper that, in the metrizable case
the situation is rather better. Indeed, we can replace the set of
-extremal points by the smaller subset of -exposed points. We
establish under general conditions on the class of functions , that every
-convex compact metrizable subset of a Hausdorff space, is the
-convex hull of its -exposed points. As a consequence we obtain
that each convex weak compact metrizable (resp. convex weak compact
metrizable) subset of a Banach space (resp. of a dual Banach space), is the
closed convex hull of its exposed points (resp. the weak closed convex hull
of its weak exposed points). This result fails in general for compact
-convex subsets that are not metrizable
On random convex analysis
Recently, based on the idea of randomizing space theory, random convex
analysis has been being developed in order to deal with the corresponding
problems in random environments such as analysis of conditional convex risk
measures and the related variational problems and optimization problems. Random
convex analysis is convex analysis over random locally convex modules. Since
random locally convex modules have the more complicated topological and
algebraic structures than ordinary locally convex spaces, establishing random
convex analysis will encounter harder mathematical challenges than classical
convex analysis so that there are still a lot of fundamentally important
unsolved problems in random convex analysis. This paper is devoted to solving
some important theoretic problems. First, we establish the inferior limit
behavior of a proper lower semicontinuous --convex function on a random
locally convex module endowed with the locally --convex topology, which
makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we
investigate the relations among continuity, locally --Lipschitzian
continuity and almost surely sequent continuity of a proper --convex
function. And then, we establish the elegant relationships among
subdifferentiability, G\^ateaux--differentiability and
Fr\'ech\'et--differentiability for a proper --convex function defined on
random normed modules. At last, based on the Ekeland's variational principle
for a proper lower semicontinuous --valued function, we show that
--subdifferentials can be approximated by subdifferentials. We
would like to emphasize that the success of this paper lies in simultaneously
considering the --topology and the locally
--convex topology for a random locally convex module.Comment: 28 page
Legendrian Realization in Convex Lefschetz Fibrations and Convex Stabilizations
In this paper, we study compact convex Lefschetz fibrations on compact convex
symplectic manifolds (i.e., Liouville domains) of dimension which are
introduced by Seidel and later also studied by McLean. By a result of
Akbulut-Arikan, the open book on , which we call \emph{convex open
book}, induced by a compact convex Lefschetz fibration on carries the
contact structure induced by the convex symplectic structure (i.e., Liouville
structure) on . Here we show that, up to a Liouville homotopy and a
deformation of compact convex Lefschetz fibrations on , any simply connected
embedded Lagrangian submanifold of a page in a convex open book on
can be assumed to be Legendrian in with the induced contact
structure. This can be thought as the extension of Giroux's Legendrian
realization (which holds for contact open books) for the case of convex open
books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one
correspondence between convex stabilizations of a convex open book and convex
stabilizations of the corresponding compact convex Lefschetz fibration. We also
show that the convex stabilization of a compact convex Lefschetz fibration on
yields a compact convex Lefschetz fibration on a Liouville domain
which is exact symplectomorphic to a \emph{positive expansion} of . In
particular, with the induced structures and are
contactomorphic.Comment: 13 pages, 1 figure, minor corrections mad
Asynchronous Convex Consensus in the Presence of Crash Faults
This paper defines a new consensus problem, convex consensus. Similar to
vector consensus [13, 20, 19], the input at each process is a d-dimensional
vector of reals (or, equivalently, a point in the d-dimensional Euclidean
space). However, for convex consensus, the output at each process is a convex
polytope contained within the convex hull of the inputs at the fault-free
processes. We explore the convex consensus problem under crash faults with
incorrect inputs, and present an asynchronous approximate convex consensus
algorithm with optimal fault tolerance that reaches consensus on an optimal
output polytope. Convex consensus can be used to solve other related problems.
For instance, a solution for convex consensus trivially yields a solution for
vector consensus. More importantly, convex consensus can potentially be used to
solve other more interesting problems, such as convex function optimization [5,
4].Comment: A version of this work is published in PODC 201
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