I can see what a convex domain and convex objective refer to in the context of graphical problems, but what about other types of problems that can't be as easily visualized? What is considered convex in those situations?

keenan

The definitions suggested here actually capture the general case quite nicely. For any vector space $X$, a set $U \subset X$ is convex if for every pair $x,y \in X$ the points $(1-t)x + t y, t \in [0,1]$ are all contained in $X$. A function $f: X \to \mathbb{R}$ is convex if for every pair $x,y \in X$, $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$. These statements make sense for any vector space; not just $\mathbb{R}^2$ or $\mathbb{R}^3$, but also for (say) any space of functions. The definitions can be extended even further by replacing the straight line segment on a flat domain (like the plane) with a straight or "geodesic" arc on curved domain (like an arc of a great circle on the sphere); here you might say that a set or function is "geodesically convex." But really, most of the basic intuition is already provided by the simple 2D examples above.

I can see what a convex domain and convex objective refer to in the context of graphical problems, but what about other types of problems that can't be as easily visualized? What is considered convex in those situations?

The definitions suggested here actually capture the general case quite nicely. For any vector space $X$, a set $U \subset X$ is convex if for every pair $x,y \in X$ the points $(1-t)x + t y, t \in [0,1]$ are all contained in $X$. A function $f: X \to \mathbb{R}$ is convex if for every pair $x,y \in X$, $f((1-t)x + ty) \leq (1-t)f(x) + tf(y)$. These statements make sense for any vector space; not just $\mathbb{R}^2$ or $\mathbb{R}^3$, but also for (say) any space of functions. The definitions can be extended even further by replacing the straight line segment on a flat domain (like the plane) with a straight or "geodesic" arc on curved domain (like an arc of a great circle on the sphere); here you might say that a set or function is "geodesically convex." But really, most of the basic intuition is already provided by the simple 2D examples above.