A subset C of an affine space or vector space such that for every pair of points x,y in C the whole line segment {(1 - t)x + t y : t ∈ [0,1]} is contained in C.

Principle

Closedness under convex combinations: if x and y belong to the set then any convex combination λx + (1−λ)y with λ in [0,1] also belongs to the set; equivalently, the set contains the straight interval between any two of its points.

Demonstration

In Euclidean space, examples include Euclidean balls, simplices, polyhedra defined as intersections of finitely many closed half-spaces, and the convex hull of any finite point set. For instance, the convex hull of three noncollinear points in R2 is the triangle whose vertices are those points.

Misapplication

Calling a star-shaped region or a simply connected region convex merely because it contains line segments from a single interior point to other points; or treating geodesic convexity on a curved manifold as ordinary convexity without qualification.

Consequence

When applied correctly, convexity yields many structural results: local minima of convex functions are global minima, separating hyperplane theorems hold in finite-dimensional spaces, extreme point characterizations (Krein–Milman in topological variants) apply, and optimization problems simplify due to uniqueness and stability properties.

Reversal

A non-convex set fails to contain some line segment between two of its points; unions or intersections that do not preserve segment inclusion illustrate the opposite behavior (for instance, two disjoint convex balls have a non-convex union).

Boundary

Refers to subsets of affine or vector spaces with the usual linear structure; does not by itself include notions such as geodesic convexity on Riemannian manifolds or convexity relative to non-linear structures unless explicitly stated.

Semantic Tension

Close terms include convex function, convex hull, and geodesic convexity; each shares the root idea of 'no inward dents' but differs in whether the object is a set, a function, or a notion dependent on a chosen path system.

Synthesis

A convex set is an entity in an affine or vector space characterized by stability under convex combinations: it contains the straight line segment between any two of its points, a property that underpins separation, extremal, and optimization phenomena in linear settings.