This chapter surveys key concepts in convex duality theory and their application to the analysis and numerical solution of problem archetypes in imaging.

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

We discuss inverse problems for the Helmholtz equation at fixed energy, specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle. In [S. Kusiak and J. Sylvester, Comm. Pure Appl. Math., 56 (2003), pp. 1525–1548], we introduced the convex scattering support of a far field, a set which will be a subset of the convex hull of the support of any source ...

Given non-degenerate intervals X of R and an increasing ordered mapping Φ : X × · · · ×X → R , games in strategic form between N players with the X as action sets with the following three properties are studied: the set of Nash equilibria E is convex, Φ is constant on E and in each Nash equilibrium at least one payoff function is not partially differentiable w.r.t. its own action. The results a...

We review the most recent developments in the area of testing by the empirical characteristic function. Our main focus is on goodness-of-fit tests based on i.i.d. observations, but we also refer to testing for symmetry and testing for independence.

Jiang, Dickey, and Kuo [12] gave the multivariate c-characteristic function and showed that it has properties similar to those of the multivariate Fourier transformation. We first give the multivariate c-characteristic function of a random functional of a Ferguson-Dirichlet process over the unit sphere. We then find out its probability density function using properties of the multivariate c-cha...

Recently Heyde, Kou and Peng (2007) proposed the notion of a natural risk statistic associated with a finite sample that relaxes the subadditivity assumption in the classical coherent risk statistics. In this note we use convex analysis to provide alternate proofs of the representation results regarding natural risk statistics.

The probability density function is a fundamental concept in statistics. Specifying the density function f of a random variable X on Ω gives a natural description of the distribution of X on the universe Ω. When it cannot be specified, an estimate of this density may be performed by using a sample of n observations independent and identically distributed (X1, ..., Xn) of X . Histogram is the ol...

We study superhedging of contingent claims with physical delivery in a discretetime market model with convex transaction costs. Our model extends Kabanov’s currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer’s robust no arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability...