Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

100%

Barov S., Dijkstra J.

Fundamenta Mathematicae

2016

tom 232

nr 2

117-129

Let 𝕍 be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim 𝕍, and let B be convex and closed in 𝕍. Let 𝓟 be a collection of linear k-subspaces of 𝕍. A set C ⊂ 𝕍 is called a 𝓟-imitation of B if B and C have identical orthogonal projections along every P ∈ 𝓟. An extremal point of B with respect to the projections under 𝓟 is a point that all closed subsets of B that are 𝓟-imitations of B have in common. A point x of B is called exposed by 𝓟 if there is a P ∈ 𝓟 such that (x+P) ∩ B = {x}. In the present paper we show that all extremal points are limits of sequences of exposed points whenever 𝓟 is open. In addition, we discuss the question whether the exposed points form a $G_{δ}$-set.

On closed sets with convex projections in Hilbert space

100%

Barov S., Dijkstra J.

Fundamenta Mathematicae

2007

tom 197

nr 1

17-33

Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set $𝓔^{k}(B)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that $𝓔^{k}(B)$ is precisely the intersection of all k-imitations C of B, i.e., closed sets C that have the same projections as B onto all k-hyperplanes. For every closed convex set B in ℓ² with nonempty interior we construct "minimal" k-imitations C, in the sense that $dim(C∖𝓔^{k}(B)) ≤ 0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Barov S.

2 Dijkstra J.

1 2016

1 2007

Wyniki wyszukiwania

On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

On closed sets with convex projections in Hilbert space