In this paper we establish a one-to-one correspondence between lawinvariant convex risk measures on L∞ and L. This proves that the canonical model space for the predominant class of law-invariant convex risk measures is L.

This article investigates the notions of exposed points and (exposed) faces in matrix convex setting. Matrix finite dimensions were first defined by Kriel 2019. Here this notion is extended to sets infinite-dimensional vector spaces. Then a connection between extreme established: point ordinary if only it exposed. leads Krein-Milman type result for that due Straszewicz-Klee classical convexity:...

A game on a convex geometry is a real-valued function de®ned on the family L of the closed sets of a closure operator which satis®es the ®nite Minkowski±Krein±Milman property. If L is the Boolean algebra 2 N then we obtain a n-person cooperative game. We will introduce convex and quasi-convex games on convex geometries and we will study some properties of the core and the Weber set of these games.

This paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions, M-convex functions and L-convex functions. Simple tie-breaking rules yield complexity bounds that are polynomials in the dimension of the variables and the size of the effective domain. Combination of the present results with a standard scaling approach leads to an efficient algor...

Abstract In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ Lp -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><m...

On the predual of a von Neumann algebra, we define a dif-ferentiable manifold structure and affine connections by embeddings into non-commutative L p –spaces. Using the geometry of uniformly convex Ba-nach spaces and duality of the L p and L q spaces for 1/p + 1/q = 1, we show that we can introduce the α-divergence, for α ∈ (−1, 1), in a similar manner as Amari in the classical case. If restric...