Mittekonvekssetes

Mittekonvekssetes are subsets of a vector space that are closed under midpoints. Formally, in a real vector space V, a subset S is mittekonvekssetes if for every x and y in S, the midpoint (x + y)/2 also lies in S. The term corresponds to the English concept of midpoint-convex sets.

Relation to convexity: A set is convex if for every t in [0,1], the combination t x + (1 − t) y belongs to the set for all x,y in S. Every convex set is mittekonvekssetes, but the converse is not true in general. Midpoint-convexity is a weaker condition, requiring only that the midpoint of any two points remains in the set.

Examples and properties: A simple example of a mittekonvekssetes that is not convex is the set of rational numbers Q inside the real line. If x,y ∈ Q, then (x+y)/2 ∈ Q, so Q is mittekonvekssetes, but choosing certain irrational t can produce irrational combinations, showing Q is not convex. The intersection of mittekonvekssetes is mittekonvekssetes, while unions of mittekonvekssetes need not preserve the property.

Regularity and consequences: In Euclidean spaces, a mittekonvekssetes that is closed is necessarily convex. The idea is that any convex combination with dyadic rational weights can be realized by repeated midpoints, and closure then ensures all convex combinations lie in the set. Similar results hold under other regularity conditions, such as local boundedness or measurability in appropriate settings.

Relation to broader concepts: Mittekonvekssetes connect to convex analysis and the study of midpoint-convex functions, providing a stepping stone from a weaker midpoint condition to full convexity under additional assumptions.