Nine convex sets determine a pentagon with convex sets as vertices

It is proved that if ℱ is a family of nine pairwise disjoint compact convex sets in the plane such that no member of ℱ is contained in the convex hull of the union of two other sets of ℱ, then ℱ has a subfamily ℱ′ with five elements such that no member of ℱ′ is contained in the convex hull of the union of the other sets of ℱ′.     

The dimension of intersections of convex sets II

The subject of this paper is a Helly type theorem which solves the following problem: Find the smallest number β= β(j,k,n) having the following property: IfG is any finite family ofβ + 1 convex sets in Euclideann-space and if the intersection of everyβ members ofG is at leastk-dimensional, then the intersection of all members ofG is at leastj-dimensional.     

Infinite dimensional convex sets with a property common to finite dimensional convex sets

We characterize those metrizable compact convex sets K that contain a Bauer simplex B such that any two affine continuous functions on K coinciding on B are equal; further, we show that all metrizable simplexes and dual balls of separable L¹-preduals have this property.     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

Extremal convex sets

Letf be an extended real valued function on the classK ⁿ of closed convex subsets of euclideann-dimensional space. A setK∈K ⁿ is said to bef-maximal if the conditionsK′∈K ⁿ ,K?K′,K≠K′ implyf(K)<f(K′), andf-minimal ifK′∈K ⁿ,K′∈K,K′≠K impliesf(K′)<f(K). In the cases whenf is the circumradius or inradius allf-maximal andf-minimal sets are determined. Under a certain regularity assumption a corresponding result is obtained for the minimal width. Moreover, a general existence theorem is established and a result concerning the existence of extremal sets with respect to packing and covering densities is proved.     

Almost convex sets

The concept of a wedgoid, a generalized wedge, is introduced and studied. Almost all convex sets are characterized by the fact that balls can be separated from such a set by means of wedgoids.     

Fixed point theorems for paracompact convex sets (II)

The present paper is the continuation of [1]. Some further generalizations of the fixed point theorems in [2] are obtained by means of the results in [1]. Project supported by the National Natural Science Foundation of China     

Empty convex polygons in almost convex sets

A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every ℓ ≥ 3, we determine the maximum size of an almost convex set that does not contain the vertex set of an empty convex ℓ-gon. Partially supported by grants T043631 and NK67867 of the Hungarian NFSR (OTKA).     

Extremality of convex sets with some applications

In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new calculus results on intersection rules for normal cones to convex sets and on infimal convolutions of support functions.     

Illuminating high-dimensional convex sets

A setL of points in thed-spaceE ^d is said toilluminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d if for every setS _i inF and every pointx in the boundary ofS _i there is a pointv inL such thatv illuminatesx, i.e. the line segment joiningv tox intersects the union of the elements ofF in exactly {x}.The problem we treat is the size of a setS needed to illuminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d . We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.     

New fixed point theorems for a family of mappings and applications to problems on sets with convex sections

Some new fixed point theorems for a family of mappings are obtained and applied to problems on sets with convex sections that were first studied by Ky Fan.