Concave and Convex Functions
Let \(X\) be a subset of a vector space over \(\mathbb{R}\) and \(f : X \to \mathbb{R}\) be a function. \(f\) is called convex if and only if for all \(t \in (0, 1)\) and \(x_{1}, x_{2} \in X\),
Intuitively, this can be thought of (in the case where \(X \subseteq \mathbb{R}\)) as the line segment joining \((x_{1}, f(x_{1}))\) and \((x_{2}, f(x_{2}))\) being above the function.
A function is called concave if the above condition is changed with the reverse inequality:
The above conditions are called strict convexity and strict concavity respectively if the inequalities are strict. This is why we use the domain \((0, 1)\) for \(t\) rather than \([0, 1]\) even though the endpoints are trivially satisfied, so that these strict cases are meaningfully defined.
An important corollary of this result gives when the value \(f(a + b)\) can be compared to \(f(a) + f(b)\) described here.