On the Geometry of M-convex Sets in the Euclidean Space

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A set A in Euclidean n-space E n , is called an m-convex set if for every m distinct points of A at least one of the line segments joining two points of them lies in A. In this article we study some geometrical and topological properties of these sets in E n .


Introduction
The notion of convexity is essential in both geometry and analysis.So, it has been generalized in many aspects and different reasons.In this article, we introduce some generalizations of this concept in E n .F.A. Valentine in [8], proved that for a closed connected 3-convex set A in E n , A is either convex or is starshaped with respect to each of its points of local non-convexity.But this is not true for a closed connected 4-convex set.So, it is not generally true for an m-convex set.Also, he proved that A can be considered as the union of three or less closed convex sets having a non-empty intersection.Also, M. Breen in [3], presented a similar decomposition without requiring the set A to be closed.It is proved in this article that, a 3-convex set A in E n is expressible as the union of at most two maximal subsets having a non-empty intersection ( kernel of A ).
The notion of radial contraction is considered.M. Beltagy and S. Shenawy in [2], proved that for a non-empty subset A of E n , p ∈ E n , and λ ∈ (0, 1), then A is convex if and only if We introduce also the concept of maffinity, and proved that C p (A) is an m-affine set if A is an m-affine set.Also, we take in consideration some geometrical and topological properties for m-convex sets such as the union of two m-convex sets, the intersection of a 2-convex set and an m-convex set, and the extreme points of an m-convex set.For more properties of m-convex sets, see [4,5].We say that x sees y via A if and only if [xy] is contained in A [2,3,5].A is called starshaped if there exists some point p in A such that p sees each point of A via A. In this case we say that A is starshaped relative to p [5].The set of all such points p is called the kernel of A, and is denoted by ker(A) [1].

Notations and Definitions
A set A ⊂ E n is called convex if for any two points p and q belonging to A, the entire segment joining p and q lies in A [6,7].The set ker(A) is convex [2].The convex hull of a set A is the smallest convex set that contains A, and is denoted by CH(A).Obviously, CH(A) = A, when A is convex [1].
Let A be a subset of the Euclidean n-space E n .The set A is said to be m-convex, m ≥ 2, if and only if for every m distinct points of A at least one of the line segments determined by these points lies in A [4,5].A maximal convex set of a set A is a convex subset of A which is not contained in another convex subset of A.
An extreme point of a set A is a point of A which is not interior to any segment with ends belonging to A. The set of all extreme points of A is denoted by E(A).For any convex set, the set remains convex if its extreme points are removed [6,7].
A hyperplane H in E n is a supporting hyperplane to A if H intersects the closure of A and A is contained in a closed side of H [7]. A supporting hyperplane H is called a locally supporting hyperplane at p ∈ A if there exists a neighborhood N of p such that N A is supported by H [7].
Let A be a non-empty subset of E n .For a fixed point p ∈ A and a fixed real number λ ∈ (0, 1), we define the λ-radial contraction of A based at p, is denoted by C λ p (A), as The set C λ p (A) is called the λ-radial contraction of A based at p.If λ = 1 2 , we get the midpoint contraction of A about p ∈ A, denoted by C p (A), and it is the set of all points 1 2 (p + x), x ∈ A, i.e., A set A is said to be affine if x, y ∈ A implies λx + (1 − λ)y ∈ A for all real numbers λ, i.e., for each pair of different points of A, the straight line L(x → y) is in A [2].A set A is called an m-affine set, m ≥ 2, if and only if for every m different points x 1 , x 2 , ..., x m in A, at least one of the straight lines L(x i → x j ) is in A, 1 ≤ i, j ≤ m.It is clear that every affine set is an maffine set.

The Results
In this section we present the main results of this paper, and we begin with the following propositions.Examples can be found to show that the converse is not generally true.

Proposition 3.2
The union of an m-convex set and a convex set is at most (m + 1)-convex set.Also, their intersection is an m-convex set.
Proof.Let A be an m-convex set, and B be a 2-convex set.First, consider (m + 1) points in A B. Now, if two points are in B say x 1 , x 2 , this implies that [x 1 x 2 ] is contained in B and hence is contained in A B. Otherwise, there exists m points in A, this implies that [ Second, suppose that A B is not an m-convex set.Then there are m points x i , 1 ≤ i ≤ m in A B such that [x i x j ] is not contained in A B for every i and j.But B is convex i.e., [x i x j ] ⊂ B for every i and j.Hence, This is a contradiction since A is an m-convex set, and the proof is complete.
For example, at m = 3, we get the intersection of a 3-convex set and a 2convex set is a 3 -convex set, as shown in Figure 1.

. It is clear that, if A is connected, then the intersection of all maximal convex subsets of A is not empty, and hence A is starshaped and ker
2.If A is not connected, we get at most (m−1) pieces and if the pieces are exactly (m − 1), then every piece is convex.The word convex is essential, as shown in Figure 3.
Proof.Suppose that p ∈ E(A), where E(A) is the set of all extreme points of A, and let x 1 , x 2 , ..., x m denote any m distinct points of B = A \ {p}.Thus x i = p for all 1 ≤ i ≤ m, and x i are points of A. Since A is an mconvex set, then at least one of the segments [x i x j ] ⊂ A, 1 ≤ i, j ≤ m.Since p ∈ E(A), i.e., p does not lie between any two points of A, and hence p does not belong to (x i x j ).Hence p does not belong to [x i x j ].Therefore, the segment Remark 3.9 Examples can be found to show that the converse of the previous theorem is not generally true.F. A. Valentine [8] proved that a closed connected 3-convex set A in E n is either convex ( has no points of local non-convexity ) or is starshaped with respect to each of its points of local non-convexity.But this is not true for a closed connected 4-convex set which is neither convex nor starshaped.Theorem 3.10 Let A be a closed connected 4-convex subset in E 2 , and let L be a straight line in the plane.Then L A consists of at most three convex sets.
Proof.Let A be a closed connected 4-convex subset in E 2 , and let L be a straight line in the plane.Suppose that L A has 4-convex sets.Let p 1 , p 2 , p 3 and p 4 be four points each of them lie in a different piece of L A, see Figure 4.It is obvious that no line segment [p i p j ] lies in L A. But all of them lie in L and so no one of them lie in A. This implies that A is not a 4-convex set which is a contradiction and the proof is complete.
Let x, y be in A. The closed segment joining x and y is denoted by [xy] where (xy) = [xy] \ {x, y} denotes the open segment joining x and y.The straight line determined by x and y is denoted by L(x → y).

Proposition 3 . 1
Let A be an m-convex subset in E n .Then A is an (m + k)-convex set for every positive integer k.

Figure 1 :
Figure 1: Intersection of a 3-convex set and a 2-convex set

Figure 4 :
Figure 4: Intersection of a straight line and a set A