On compact convex sets with given extreme points

The research was financially supported by the Russian Foundation for Fundamental Research (Grant 94-01-01286).     

On extreme points of bounded sets of generalized finite sequence spaces

The Linear Independence Extreme Point Theorem and Opposite Sign Theorem of semiinfinite programming provide algebraic characterizations of unbounded infinite dimensional convex polyhedral sets in terms of their extreme points. The Charnes, Kortanek, and Raike purification algorithm, which transforms any non-extreme point to an extreme point having objective function value at least as great, is based upon the constructive proofs of these two theorems. In this paper these extreme point characterizations are extended to additionally constrained, bounded convex polyhedral constraint sets of semi-infinite programming. It is shown that each of these bounded sets is spanned by a possibly infinite number of extreme points, and as before the proofs are constructive, thereby expanding the range of applicability of the purification algorithm to these cases.     

On farthest points of sets

A set K in a normed linear space is said to be M-compact if any maximizing sequence in K is compact. A sequence {g_n} in K is called maximizing if for some x?X, {∥ x ? g_n ∥} converges to the farthest distance between x and K. In this paper we study M-compact sets, relate the continuity behavior of the associated farthest-point map with the ?ateaux differentiability of the farthest-distance function, and prove that in a normed space admitting centers any nonempty M-compact set having the unique farthest-point property must be a singleton.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

On extreme points inl 1

It is proved that every bounded closed and convex subset ofl ₁ is the closed convex hull of its extreme points. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

The extreme points of some convex sets in the theory of majorization

Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ω_{µ, w}⁺f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩ_{µ, w} ⁺f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given.     

On distance sets of large sets of integer points

Distance sets of large sets of integer points are studied in dimensions at least 5. To any ε > 0 a positive integer Q _ɛ is constructed with the following property; If A is any set of integer points of upper density at least ε, then all large multiples of Q _ɛ ² occur as squares of distances between the points of the set A. Research supported in part by NSF Grant DMS-0456490.     

On finite sets of points inP 3

Given a finite set of points Γ₀ which span a projective spaceP ³, we show here that a plane spanned by points of Γ₀ can be a neighbour of at most eight points of Γ₀, these being the vertices of a projective cube; the common neighbour plane is then elementary with the three only points of Γ₀ in it being diagonal points of the cube. This extends toP ³ some results of L. M. Kelly and W. O. J. Moser in the planeP ².     

On extreme points in separable conjugate spaces

It is proved that every bounded closed and convex subset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points.     

On the index of approximating sets of periodic points

Summary LetK be a compact space andf:K→K a continuous map without fixed points, i.e. Fixf=⊘. For prime numbersp, the sets Fixf ^p are freeℤ/p-spaces with theℤ/p-action induced byf. Our aim is to estimate the topological indicesi(F _p,f) of invariant subsetsF _p⊂Fixf ^p approximating a givenS⊂K. We construct an example (K,f,S) withS⊂Fixf ^q (q being some prime number) such that, for each neighborhoodU ofS, i (Fix (f|u) ^p, f) increases linearly withp. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.