In this paper, we prove the following results: Let K be a closed convex subset of a real Banach space E. Let T := {T (t) | t ∈ R+} be strongly continuous semigroup of asymptotically nonexpansive mappings from K into K such that F (T ) := ∩t∈R+F (T (t)) 6= ∅, where F (T (t)) = {x ∈ K |T (t)x = x} and R+ denotes the set of nonnegative real numbers. Then for arbitrary x0 ∈ K, the implicit iteratio...

The authors exhibit an abelian group not embeddable in a product of copies of Z (the integers) but totally orderable so that every covering pair of convex subgroups has archimedean quotient isomorphic to Z (= g.d.), thus furnishing a counterexample to Théorème 4 on p. 53 of [21]; then go on to show that every countable g.d. group is algebraically free (§4); and purport to show it also embeddabl...

The definition of n-width of a bounded subset A in a normed linear space X is based on the existence of n-dimensional subspaces. Although the concept of an n-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of n-widths for a metric tree, called Tn-widths. Later we discuss properties of Tn-widths, and s...

The aim of this paper is to motivate the development of a Brunn-Minkowski theory for minimal surfaces. In 1988, H. Rosenberg and E. Toubiana studied a sum operation for finite total curvature complete minimal surfaces in R3 and noticed that minimal hedgehogs of R3 constitute a real vector space [14]. In 1996, the author noticed that the square root of the area of minimal hedgehogs of R3 that ar...

In every dimension d ≥ 1, we establish the existence of a positive finite constant vd and of a subset Ud of R d such that the following holds: C + Ud = R d for every convex set C ⊂ R of volume at least vd and Ud contains at most log(r) r points at distance at most r from the origin, for every large r.

Let F be the finite field in q = p1 elements, £(x) be a A:-tuple of polynomials in F [xx,..., x„], V be the set of points in ¥'j satisfying F(x) = 0 and S,T be any subsets of F;. Set y)= L e{— TrU '?)) íory±Q, and 4>(K) = maxv\ 02( V)q2k. In case q = p v/e deduce from this, for example,...

In a related paper [2], the authors have shown that a homeomorphism which preserves convex sets, mapping an open subset of one locally convex topological vector space onto an open subset of another, is a projective map (the quotient of an affine operator by an affine functional). The establishment of this result in its full generality required a treatment of (possibly infinite dimensional) topo...

The minimum of quadratic functionals of the gradient on the set of convex functions ABSTRACT We study the innmum of functionals of the form R Mruru among all convex functions u 2 H 1 0 (() such that R jruj 2 = 1. (is a convex open subset of R N , and M is a given symmetric N N matrix.) We prove that this innmum is the smallest eigenvalue of M if is C 1. Otherwise the picture is more complicated...

Let K be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self-mapping A of K , we denote by Lip(A) its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self-mappings of K . We consider the set of all sequences {At}t=1 of such selfmappings with the property limsupt→∞ Lip(At) ≤ 1. Endowing it with an appropriate...

Let D be a nonempty compact subset of a Banach space X and denote by S(X) the family of all nonempty bounded closed convex subsets of X. We endow S(X) with the Hausdorff metric and show that there exists a set F ⊂ S(X) such that its complement S(X) \ F is σ-porous and such that for each A ∈ F and each x̃ ∈ D, the set of solutions of the best approximation problem ‖x̃− z‖ → min, z ∈ A, is nonempty...