Convex and discrete geometry

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• ConceptConvex set

  A set of points is convex if it contains the whole segment between each two of its points. A solid ball and a solid cube are convex; a crescent is not.

• ConceptConvex hull

  The convex hull of a set of points is the smallest convex set that contains it. It is the set of all convex combinations of points of the set.

• ConceptConvex combination

  A convex combination of finitely many points is a weighted sum of the points whose weights are nonnegative and add up to $1$. It is the special kind of affine combination in which no weight is negative. The convex hull of a set is exactly the set of all convex combinations of its points. A convex set is a set closed under convex combinations. So convex combinations are the basic operation of convex geometry.

• ConceptLine segment

  The line segment between two points is the set of all points on the straight path from one to the other, endpoints included. In a real vector space it is the set of convex combinations of the two points. A set is convex exactly when it contains the segment between any two of its points, so segments are the building blocks of convexity.

• ConceptOpen segment

  The open segment between two points is the line segment between them with the two endpoints removed, when the points are distinct. Open segments appear in the definitions of extreme points, extreme sets and faces. These notions ask what happens when a point of a set lies strictly between two other points. They also define strict convexity of sets.

• ConceptConvex join

  The convex join of two sets is the union of all line segments that start in the first set and end in the second. When both sets are convex and nonempty, the convex join is the convex hull of their union. So it gives a concrete way to build the hull of two convex pieces from segments alone.