On criteria of Blumenthal for inner-product spaces
On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space
Let be a Euclidean space or the Hilbert space ℓ², let k ∈ ℕ with k < dim , and let B be convex and closed in . Let be a collection of linear k-subspaces of . A set C ⊂ is called a -imitation of B if B and C have identical orthogonal projections along every P ∈ . An extremal point of B with respect to the projections under is a point that all closed subsets of B that are -imitations of B have in common. A point x of B is called exposed by if there is a P ∈ such that (x+P) ∩ B = x. In the present...
On prior distributions which give rise to a dominated Bayesian experiment.
On reduced pairs of bounded closed convex sets.
In this paper certain criteria for reduced pairs of bounded closed convex set are presented. Some examples of reduced and not reduced pairs are enclosed.
On the number of minimal pairs of compact convex sets that are not translates of one another
Let [A,B] be the family of pairs of compact convex sets equivalent to (A,B). We prove that the cardinality of the set of minimal pairs in [A,B] that are not translates of one another is either 1 or greater than ℵ₀.
On the reduction of pairs of bounded closed convex sets
Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ C. We prove two main theorems on reduction of equivalent pairs. The first theorem implies that, in a finite-dimensional space, a pair of nonempty compact convex sets with a piecewise smooth boundary and parallel tangent spaces at some boundary points is not minimal. The second theorem...
On the size of approximately convex sets in normed spaces
Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
On the structure of support point sets.
Pairs of sets with convex union.
In this paper the notion of convex pairs of convex bounded subsets of a Hausdorff topological vector space is introduced. Criteria of convexity pair are proved.
Parallel methods in image recovery by projections onto convex sets
Points extrêmaux dans un monoide affine semitopologique.
Produit tensoriel de deux cônes convexes saillants
Reflexivity of convex subsets of L(H) and subspaces of .
Remarks on independent sequences and dimension in topological linear spaces.
Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces
For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class...
Separation by hyperplanes in finite-dimensional vector spaces over Archimedean ordered fields.
Sets invariant under projections onto one dimensional subspaces
The Hahn–Banach theorem implies that if is a one dimensional subspace of a t.v.s. , and is a circled convex body in , there is a continuous linear projection onto with . We determine the sets which have the property of being invariant under projections onto lines through subject to a weak boundedness type requirement.
Sets of constant width and diametrically complete setes in normed spaces
Smoothness of vector sums of plane convex sets.
Some notes on convex sets.