Convex Set
From the book
Chapter 12: Optimization Methods. In the chapter mind map this icon labels Convex Sets & Functions. The discussion below is excerpted and lightly edited from § Convex Sets in Mathematics for AI and Machine Learning.
A set $S$ is convex if for any $x, y \in S$ and any $\lambda \in (0,1)$,
the book figure illustrates the geometric definition of convexity. The left panel shows a convex set (ellipse) where the line segment between any two points $x$ and $y$ (red dashed line) remains entirely inside the set. This demonstrates that for any $\lambda \in (0,1)$, the point $\lambda x + (1-\lambda)y$ lies within the set. The right panel shows a non-convex set (crescent shape) where the line segment between two points crosses the boundary, violating the convexity condition. This visualization makes concrete the definition: a set is convex if and only if it contains all line segments connecting any pair of its points. This geometric property is fundamental to convex optimization, as it ensures that any local minimum in a convex set is also a global minimum.
What this drawing shows
What you see. Shows a set where every segment between two points remains inside the set.
In the mind map. Chapter 12 — Convex Sets & Functions. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 12: Optimization Methods — see the mind-map node Convex Sets & Functions.
