In this paper we provide a rigorous toolkit for extending convex risk measures from L∞ to L, for p ≥ 1. Our main result is a one-to-one correspondence between law-invariant 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. Some significant counterexamples illustrate the many pitfalls with convex ris...

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let {Ti}i=1 be N nonexpansive self-mappings of K with F = ∩i=1F (Ti) ̸= ∅ (here F (Ti) denotes the set of fixed points of Ti). Suppose that one of the mappings in {Ti}i=1 is semi-compact. Let {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1) and {βn} ⊂ [τ, 1] for some τ ∈ (0, 1]. For arbitrary x0 ∈ K, let the sequence {x...

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the ǫ-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C. 1

The aim of this paper is to present some existence results for nearest-point and farthestpoint problems, in connection with some geometric properties of Banach spaces. The idea goes back to Efimov and Stečkin who, in a series of papers (see [28, 29, 30, 31]), realized for the first time that some geometric properties of Banach spaces, such as strict convexity, uniform convexity, reflexivity, an...

Let E be a subset of a convex, open, bounded, planar set G. Let P (E, G) be the relative perimeter of E (the length of the boundary of E contained in G). We obtain relative geometric inequalities comparing the relative perimeter of E with the relative diameter of E and with its relative inradius. We prove the existence of both extremal sets and maximizers for these inequalities and describe the...

In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector space R and vectorvalued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞ and inequalities when these vector-valued functions are in a weakly* compact subset of a Banach vector space gener...

Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. We prove (under certain assumptions on f) that the iterates of a generic mapping in these spaces converge strongly to a retract...

In this paper we deal with the following generalized vector quasiequilibrium problem: given a closed convex set K in a normed space X, a subset D in a Hausdorff topological vector space Y , and a closed convex cone C in R. Let Γ : K → 2, Φ : K → 2 be two multifunctions and f : K×D×K → R be a single-valued mapping. Find a point (x̂, ŷ) ∈ K×D such that (x̂, ŷ) ∈ Γ(x̂)× Φ(x̂), and {f(x̂, ŷ, z) : z ∈ Γ(...

CONVEXITY RELATED ISSUES FOR THE SET OF HYPERGRAPHIC SEQUENCES Hasmik Sahakyan, Levon Aslanyan Abstract: We consider ( ), the set of all degree sequences of simple hypergraphs with vertices and hyperedges. We show that ( ), which is a subset of the -dimensional + 1-valued grid , is not a convex subset of ; and give a characterization of the convex hull of ( ).

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.