By using parameterized representation of fuzzy numbers, criteria for a lower semicontinuous fuzzy mapping defined on a non-empty convex subset of Rn to be a convex fuzzy mapping are obtained. c © 2007 Elsevier Ltd. All rights reserved.

We prove that any locally bounded from below, upper semicontinuous v-convex function in any Carnot group is h-convex. §

We introduce semicontinuity concepts for functions f with values in the space C(Y ) of closed convex subsets of a finite dimensional normed vector space Y by appropriate notions of upper and lower limits. We characterize the upper semicontinuity of f : X → C(Y ) by the upper semicontinuity of the scalarizations σf( · )(y∗) : X → R by the support function. Furthermore, we compare our semicontinu...

In this paper, we give a fundamental convexity preserving for spectral functions. Indeed, we investigate infimal convolution, Moreau envelope and proximal average for convex spectral functions, and show that this properties are inherited from the properties of its corresponding convex function. This results have many applications in Applied Mathematics such as semi-definite programmings and eng...

In this paper we prove that every proper convex and lower semicontinuous functional de ned on a real re exive Banach space X is semicoercive if and only if every small uniform perturbation of attains its minimum value on X.

It is shown that the integral functional I(y,z) = J"0 f(t,y(t), z(t))dp. is lower semicontinuous on its domain with respect to the joint strong convergence of yk -» y in Lp(G) and the weak convergence of zk -» z in LAG), where 1 < p < oo and 1 < q < oo, under the following conditions. The function/: (t,x,w) -*f(t,x,w) is measurable in / for fixed (x, w), is continuous in (x, w) for a.e. /, and ...

We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completion K̄, a convex function Kn → K̄ which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation...

For a set S in a Banach space, we denote by dim(S) its covering dimension [1, p. 42]. Recall that, when S is a convex set, the covering dimension of S coincides with the algebraic dimension of S, this latter being understood as ∞ if it is not finite [1, p. 57]. Also, S and conv(S) will denote the closure and the convex hull of S, respectively. In [3], we proved what follows. dim({x ∈ X : Φ(x) =...