Definition

A set is convex if for any , any , we have

How to judge whether a set is convex

Way 1

The first way is to use the Definition![[Convexity#Definition]]

Way 2

The second way is that a set is convex if and only if the intersection intersection with an arbitrary line is convex. That is to say, if we let and let we would have which becomes a one-variable function. We could only consider the convexity of .

Note: The here represents the starting point of a line and the vector represents the direction of the line

本质就是构造一个直线和原来的相交，证明是一个凸函数，把函数变成一个单变量的函数

Related

[[Operations that preserve convexity]]
[[Epigraph]]
[[Sublevel and superlevel set]]