We establish the following max-plus analogue of Minkowski’s theorem. Any point of a compact max-plus convex subset of (R∪{−∞})n can be written as the max-plus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set ca...

In 1906 Jensen founded the theory of convex functions. This enabled him to prove a considerable extension of the AM-GM inequality. Recall that a subset D of a real vector space is called convex if every convex linear combination of a pair of points of D is in D. Equivalently, if x, y ∈ D, then tx+ (1− t)y ∈ D for every t ∈ [0, 1]. Given a convex set D, a function f : D → R is called convex if f...

*Correspondence: [email protected]; [email protected] Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China Abstract Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let {Tn} : C→ C be a countable family of weak relatively nonexpansive mappings such that F = ⋂∞ n=1 F(Tn) = ∅. For any given gauss x0 ∈ C,...

and Applied Analysis3It is clear that every k-Lipschitz mapping is continuous. Moreover, the Banach contrac-tion principle holds for a closed subset in a complete p-normed space.Definition 2.2 see 3 . Let X, ‖ · ‖p 0 < p ≤ 1 be a p-normed space and 0 < s ≤ p. A setC ⊂ X is said to be s-convex if the following condition is satisfied1 − t x t1/sy ∈ C, whenever x, y ∈ C, t ...

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A ∈ K(X) : A∩G = ∅}. We prove that the set of all A ∈ KG(X) (resp. A ∈ K(X)), such that the minimizat...

and Applied Analysis 3 Lemma 3. Let p > 1 and E be a p-uniformly convex and smooth Banach space. Then, for each x, y ∈ E, φ p (x, y) ≥ c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p (8) holds, where c 0 is maximum in Remark 2. Proof. Let x, y ∈ E. By Theorem 1, we have ‖x‖ p ≥ 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p + p⟨x − y, J p y⟩ + c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p , (9) where c 0 is maximum in Remark 2. Hence, we get φ p (x, y) = ‖x‖ p − 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p − p...

In the unit disk D hyperbolic geodesic rays emanating from the origin and hyperbolic disks centered at the origin exhibit simple geometric properties. The goal is to determine whether analogs of these geometric properties remain valid for hyperbolic geodesic rays and hyperbolic disks in a simply connected region Ω. According to whether the simply connected region Ω is a subset of the unit disk ...

In a recent preprint by Amaral & Letchford (2006) convex hulls of sets of matrices corresponding to permutations and path-metrics are studied. A symmetric n × n-matrix is a path metric, if there exist points x1, . . . , xn ∈ R such that the matrix entries are just the pairwise distances |xk − xl| between the points and if these distances are at least one whenever k 6= l. The convex hull of the ...

Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by {Tj : C → C}j=1 as follows: Un(j) = an(j)I+bn(j)T j Un(j−1)+cn(j)un(j), j = 1,2, . . . ,r , x1 ∈ C , xn+1 = an(r)xn+bn(r)T r Un(r−1)xn+cn(r)un(r), n≥ 1, where Un(0) := I, I the identity map; and {un(j)} are bounded s...

For a real vector space V acted on by a group K and fixed x and y in V , we consider the problem of finding the minimum (resp., maximum) distance, relative to a Kinvariant convex function on V , between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V . Then we use this result to ...