### A bipolar theorem for L${}_{+}^{0}(\Omega ,\mathcal{F},\mathbf{P})$

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We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series $g$ is convex and balanced which implies that the subset ${\overline{\mathbb{X}}}_{g}$ consisting of formal power series which can be composed by a formal power series $g$ possesses such properties. We also provide a necessary and sufficient condition for the superposition operator ${T}_{g}$ to map ${\overline{\mathbb{X}}}_{g}$ into itself or to map ${\mathbb{X}}_{g}$ into...

This paper gives necessary and sufficient conditions for the closure of a face in a compact convex set to be again a face. As applications of these results, several theorems scattered in the literature are proved in an economical and uniform manner.