Mathematical Notes, vol. 71, no. 1, 2002, pp. 34–38. Translated from Matematicheskie Zametki, vol. 71, no. 1, 2002, pp. 37–42. Original Russian Text Copyright c 2002 by M. V. Balashov. An Analog of the Krein–Mil  man Theorem for Strongly Convex Hulls in Hilbert Space M. V. Balashov Received May 15, 2000 Abstract—We prove the following theorem: in Hilbert space a closed bounded set is contained in the strongly convex R-hull of its R-strong extreme points. R-strong extreme points are a subset of the set of extreme points (it may happen that these two sets do not coincide); the strongly convex R-hull of a set contains the closure of the convex hull of the set. Key words: Hilbert space, convexity, extreme point, strongly extreme point, R-exposed point, strongly convex hull. 1. INTRODUCTION Suppose that H is a Hilbert space over the field of real scalars. By [x, y] we denote the closed interval with endpoints x and y . By B r (a) we denote the ball of radius r centered at the point a , i.e., B r (a)= {x ∈H|x − a≤ r} , ∂B r (a)= {x ∈H|x − a = r} . By a, b we denote the inner product of the vectors a, b ∈H . For a set A ⊂H and a point x , we introduce ρ(x, A) = inf y∈A x − y . Let P A be the projection operator onto a convex closed set A . The Hausdorff distance between the sets A and B is defined by h(A, B) = inf {r> 0 | A ⊂ B + B r (0) , B ⊂ A + B r (0)}. Let us also introduce some useful notation: for q ∈ ∂B 1 (0) and a ∈H , let l(a, q)= {a + λq | λ ≥ 0} , ∂B r (a, q)= {x ∈H|x − a = r, q,x − a≥ 0} , i.e., this is the “upper” (in the q-direction) hemisphere of the sphere ∂B r (a). Let us now state some definitions and well-known results. Definition 1.1 [1, 2]. For a set A ⊂H , by a strongly convex R-hull we mean the set formed by the intersection of all balls of radius R containing A . Denote this set by strco R A . Definition 1.2 [1, 2]. For a set A ⊂H , a point x ∈ A is said to be R-strongly extreme if for every pair of points y,z ∈ A: y = x , z = x , we have x/ ∈ strco R {y,z} . Denote the set of R-strongly extreme points of A by extr R A . Definition 1.3 [2]. For a set A ⊂H , a point x ∈ A is said to be R-exposed if there exists a ball of radius R centered at the point z such that A ⊂ B R (z) and A ∩ ∂B R (z)= {x} . Denote the set of R-exposed points of A by exp R A . Let extr A be the subset of extreme points of the set A . Note that if strco R A ∩ extr R A = ∅ , then extr R A ⊂ extr A and coA ⊂ strco R A . In [2] the following theorem was proved. 34 0001-4346/2002/7112-0034$27.00 c 2002 Plenum Publishing Corporation