Support functions of general convex sets

No Abstract. .In Memory of Gian-Carlo Rota     

A general theory of tensor products of convex sets in Euclidean spaces

Positivity - We introduce both the notions of tensor product of convex bodies that contain zero in the interior, and of tensor product of 0-symmetric convex bodies in Euclidean spaces. We prove...     

Some properties of convex hulls of integer points contained in general convex sets

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.     

A characterization of convex calibrable sets in

The main purpose of this paper is to characterize the calibrability of bounded convex sets in by the mean curvature of its boundary, extending the known analogous result in dimension 2. As a by-product of our analysis we prove that any bounded convex set C of class C^1,1 has a convex calibrable set K in its interior, and and for any volume V [|K|,|C|] the solution of the perimeter minimizing problem with fixed volume V in the class of sets contained in C is a convex set. As a consequence we describe the evolution of convex sets in by the minimizing total variation flow.Mathematics Subject Classification (2000): 35J70, 49J40, 52A20, 35K65     

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).     

A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection P _n of points on the plane contains two such that any circle containing them contains n/c elements of P _n , c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S _i , S _j of Φ such that any set S' homothetic to S that contains them contains n/c elements of Φ, c a constant (S' is homothetic to S if , where λ is a real number greater than 0 and is a vector of ). Our proof method is based on a new type of Voronoi diagram, called the ``closest covered set diagram' based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in such that for any nonempty subset Φ _H of Φ there is a sphere S _H containing all the elements of Φ _H , and no other element of Φ. Received December 7, 1994, and in revised form March 25, 1996.     

A representation of convex semilinear sets

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Vélez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Vélez.     

A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

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.     

On the length of longest alternating paths for multicoloured point sets in convex position

Let P be a set of points in R² in general position such that each point is coloured with one of k colours. An alternating path of P is a simple polygonal whose edges are straight line segments joining pairs of elements of P with different colours. In this paper we prove the following: suppose that each colour class has cardinality s and P is the set of vertices of a convex polygon. Then P always has an alternating path with at least (k-1)s elements. Our bound is asymptotically sharp for odd values of k.     

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′.     

A remark on intersection of convex sets

Let X be a real Banach space. Let be a family of closed, convex subsets of X. We show that either the intersection ?_γ∈Γ(G_γ)^? of the ?-neighborhood of the sets G_γ is bounded for each ?>0, or it is unbounded for each ?>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces.     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that