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 ${}^k\left(B\right)$ consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that ${}^k\left(B\right)$ is precisely the intersection of all k-imitations...

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...

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.

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.

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...

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 $\mathscr{H}(A,Co\left(A\right))\ge lo{g}_{2}n-1$ and $diam\left(A\right)\le C\surd n{\left(lnn\right)}^2$, 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.