Dual Extremal Problem Research Articles

Area-perimeter duality in polygon spaces

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.

Structure of optimal martingale transport plans in general dimensions

Given two probability measures $\mu$ and $\nu$ in “convex order” on $\mathbb{R}^{d}$, we study the profile of one-step martingale plans $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\mathbb{R}^{d}$ is transported by the optimal martingale plan into a probability measure $\pi_{x}$ concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost $c(x,y)=\vert x-y\vert $ in the two-dimensional case, and also for any $d\geq3$ as long as the marginals are in “subharmonic order.” In some cases, $\pi_{x}$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that “martingale supporting” Borel subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ can be decomposed into a collection of mutually disjoint components by means of a “convex paving” of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain “optimal martingale supporting” Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.

Recovering Linear Operators and Lagrange Function Minimality Condition

This article concerns the recovery of the operators by noisy information in the case that their norms are defined by integrals over infinite intervals. We study the conditions under which the dual extremal problem (often nonconvex) can be solved using the Lagrange function minimality condition.

An extremal problem for characteristic functions

Suppose E E is a subset of the unit circle T \mathbb {T} and H ∞ ⊂ L ∞ H^\infty \subset L^\infty is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E E to z n H ∞ z^nH^\infty . This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

Optimal recovery of operators and multidimensional Carlson type inequalities

The paper is concerned with recovery problems of linear multiplier-type operators from noisy information on weighted classes of functions. Optimal methods of recovery are constructed. The dual extremal problem is closely connected with Carlson type inequalities.

The extremal function in the Delsarte problem of finding an upper bound for the kissing number in the three-dimensional space

We consider an extremal problem for continuous functions that are nonpositive on a closed interval and can be represented by series in Legendre polynomials with nonnegative coefficients. This problem arises from the Delsarte method of finding an upper bound for the kissing number in the three-dimensional Euclidean space. We prove that the problem has a unique solution, which is a polynomial of degree 27. This polynomial is a linear combination of Legendre polynomials of degrees 0, 1, 2, 3, 4, 5, 8, 9, 10, 20, and 27 with positive coefficients; it has simple root 1/2 and five double roots in (−1, 1/2). We also consider the dual extremal problem for nonnegative measures on [−1, 1/2] and, in particular, prove that an extremal measure is unique.

The Carathéodory and Kobayashi/Royden metrics by way of dual extremal problems

We study the Carathéodory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly, incisive results are obtained for convex domains.

Extremal problems for quasiconformal maps of punctured plane domains

The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.

On optimal recovery methods in Hardy-Sobolev spaces

A general approach to the construction of optimal methods of recovery of linear functionals from a known solution of the dual extremal problem is proposed which is based on a certain parametrization of this solution of the dual problem. Using this approach several optimal recovery problems in Hardy-Sobolev classes are successfully solved, including the recovery of functions from information about their Fourier coefficients or about the values of the function at some system of nodes, in the periodic and non-periodic cases.

On some extremal problems in H p and the prediction ofL p -harmonizable stochastic processes

We providesimple andsuccinct solutions to two dual extremal problems in the Hardy spacesHp, and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter “Lp-harmonizable” stochastic processes, for all 1≦p<∞. Two of the results presented appear new. The methods of proof of the rest of the results provide alternatesimpler andshorter proofs for some earlier known theorems.

Simplification of dual extremal problems invariant with respect to change in variables

Simplification of dual extremal problems invariant with respect to change in variables

Smoothing and Interpolation in a Convex Subset of a Hilbert Space

Interpolation and smoothing subject to convex constraints is considered. We clarify instances in a Hilbert space when the problem of finding the least norm solution to these problems can be separated into first finding the orthogonal projection onto the constraint set and then fitting the interpolation or smoothing requirements by solving a finite dimensional dual extremal problem. We report on some numerical experiments with this approach for a problem in EXAFS spectroscopy and also relate our results to special cases studied by others.