Convessità

Convessità is a mathematical property of functions and sets that describes the shape of their graphs or boundaries. In the context of functions, a function is said to be convex if the line segment joining any two points on its graph lies entirely above or on the graph. This property is formally defined using inequalities: for a function *f* defined on an interval, it is convex if for all *x* and *y* in the domain and all *λ* between 0 and 1, the following holds:

*f(λx + (1−λ)y) ≤ λf(x) + (1−λ)f(y)*

This inequality is known as Jensen’s inequality, and it implies that the function lies below its secant lines. Convex functions are widely used in optimization problems due to their well-behaved nature, such as having a single global minimum.

In geometry, a set is convex if, for any two points within the set, the line segment connecting them is entirely contained within the set. This concept extends to higher dimensions, where convex sets include shapes like balls, cubes, and pyramids. Non-convex sets, such as crescents or star-shaped figures, have regions where line segments between points fall outside the set.

Convexity plays a crucial role in various fields, including economics (e.g., production possibility frontiers), operations research (e.g., linear programming), and machine learning (e.g., optimization algorithms). The dual concept, concavity, describes functions or sets where the line segment between any two points lies entirely below or on the graph or boundary, often used in contexts like risk assessment or utility theory. The study of convexity is foundational in real analysis, functional analysis, and computational mathematics.