Hahn Banach Separation Theorem Research Articles

AN ANALOGY OF HAHN–BANACH SEPARATION THEOREM FOR NEARLY TOPOLOGICAL LINEAR SPACES

In this paper, we introduce the notion of nearly topological linear spaces and use it to formulate an alternative definition of the Hahn–Banach separation theorem. We also give an example of a topological linear space to which the result is not valid. It is shown that \(\mathbb{R}\) with its ordinary topology is not a nearly topological linear space.

Noncommutative Partial Convexity Via $$\Gamma $$-Convexity

Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix polynomials with constrained terms. Given a tuple of symmetric polynomials $\Gamma$, a free set is called $\Gamma$-convex if it closed under isometric conjugation by isometries intertwining $\Gamma$. We establish an Effros-Winkler Hahn-Banach separation theorem for $\Gamma$-convex sets; they are delineated by linear pencils in the coordinates of $\Gamma$ and the variables $x$.

A second welfare theorem in a non-convex economy: The case of antichain-convexity

We introduce the notion of an antichain-convex set to extend Debreu (1954)’s version of the second welfare theorem to economies where either the aggregate production set or preference relations are not convex. We show that – possibly after some redistribution of individuals’ wealth – the Pareto optima of some economies which are marked by certain types of non-convexities can be spontaneously obtained as valuation quasiequilibria and equilibria: both equilibrium notions are to be understood in Debreu (1954)’s sense. From a purely structural point of view, the mathematical contribution of this work is the study of the conditions that guarantee the convexity of the Minkowski sum of finitely many possibly non-convex sets. Such a study allows us to obtain a version of the Minkowski\\Hahn–Banach separation theorem which dispenses with the convexity of the sets to be separated and which can be naturally applied in standard proofs of the second welfare theorem; in addition – and equally importantly – the study allows to get a deeper understanding of the conditions on the single production sets of an economy that guarantee the convexity of their aggregate.

Circular free spectrahedra

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by eit: up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials.

General financial market model defined by a liquidation value process

Financial market models defined by a liquidation value process generalize the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. The solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex, e.g. in presence of proportional transaction costs and fixed costs. Therefore, the classical duality principle based on the Hahn–Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. Using an alternative method based on the concepts of essential supremum and maximum, we provide a characterization of European and American contingent claim prices under the absence of arbitrage opportunity of the second kind.

Corrigendum to: Operator monotone functions and Löwner functions of several variables

We fix a gap in the proof of Theorem 7.24 in Ann. of Math. 176 (2012), 1783–1826. There is a gap in the proof of Theorem 7.24 in [1], though the statement of the theorem is correct. In the proof of necessity, we argue that Λ is in G by contradiction. If it were not, invoking the Hahn-Banach separation theorem would yield a real skew-symmetric matrix K and a constant δ ≥ 0 such that tr(ΓK) ≥ −δ for all Γ in G, and tr(ΛK) < −δ. In the proof we assumed that δ = 0, but this assumption is unjustified. Instead, we argue as follows. Define ∆ by ∆ij = (x r j − xi )Kji, i 6= j, and with the diagonal entries ∆ii chosen so that each ∆ r ≥ 0 and so that

General Financial Market Model Defined by a Liquidation Value Process

We present a general financial market model defined by a liquidation value process. This approach generalizes the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. This allows to consider financial market models where proportional transaction costs and fixed costs coexist. In this case, the solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex. Therefore, the usual duality principle based on the Hahn-Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. We propose an alternative method to price European or American contingent claims under absence of arbitrage opportunity of the second kind.

A nonconvex separation property and some applications

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multipliers in Optimization.

A nonconvex separation property in Banach spaces

We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our property with the extremal principle of Mordukhovich.