In [K.-I. Mitani and K.-S. Saito, J. Math. Anal. Appl. 327 (2007), 898–907] we characterized the strict convexity, uniform convexity and uniform non-squareness of Banach spaces using ψ-direct sums of two Banach spaces, where ψ is a continuous convex function with some appropriate conditions on [0, 1]. In this paper, we characterize the Bn-convexity and Jn-convexity of Banach spaces using ψ-dire...

Let C be a bounded closed convex subset of a uniformly convex Banach space X and let = = {T (t) : t ∈ G} be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we provide the strong mean ergodic convergence theorem for the almost-orbit of =.

Given x0, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any convex function f : C → R is upper semi-continuous at x0, and (ii) C is polyhedral at x0. In the particular setting of closed convex mappings and Fσ domains, we prove that any closed convex function f : C → R is continuous at x0 if and only if C is polyhedral at x0. Th...

Let S be a finite set of n points in the plane in general position. A k-hole of S is a simple polygon with k vertices from S and no points of S in its interior. A simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. Moreover, a point set S is l-convex if there exists an l-convex polygonalization of S. Considering a typical Erdős-Szek...

The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of this graph for codimen-sion one subgroups leads to splittings. We also reprov...

Let k be a field, C/k a proper, smooth, geometrically connected curve of genus g ≥ 1 given with a marked rational point 0 ∈ C(k), JC/k := Pic 0 C/k its Jacobian. Concretely, the group JC(k) is the group (under tensor product) of isomorphism classes of invertible sheaves L on C of degree zero. Given a point P ∈ C(k), we denote by I(P ) ⊂ OC the ideal sheaf of functions vanishing at P . Given P1,...

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...

The notion of an asymptotic center is used to prove a number of results concerning the existence of fixed points under certain selfmappings of a closed and bounded convex subset of a uniformly convex Banach space.

Definition 1.1. Let C be a subset of R. We say C is convex if αx+ (1− α)y ∈ C, ∀x, y ∈ C, ∀α ∈ [0, 1]. Definition 1.2. Let C be a convex subset of R. A function f : C 7→ R is called convex if f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y), ∀x, y ∈ C, ∀α ∈ [0, 1]. The function f is called concave if −f is convex. The function f is called strictly convex if the above inequality is strict for all x, y ∈ C wi...

Let B (resp. K , BC , K C ) denote the set of all nonempty bounded (resp. compact, bounded convex, compact convex) closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B stand for the set of all F ∈ B such that the problem (F,G) is well-posed. We proved that, if X is strictly convex and Kadec, the set ...