convex

A convex function $f$ satisfies the property that a line between any two points on its graph is on or above the graph:

for any $t \in [0, 1]$. It is strictly convex if this inequality is strict. Wikipedia has this nice illustration of the definition:

A slight generalization of this definition, $E[f(x)] \ge f(Ex)$, is sometimes called Jensen's inequality.

The second derivative of a differentiable convex function must be nonnegative; that is, the derivative must be non-decreasing. In the multivariate setting, the Hessian of a differentiable convex function must be positive semidefinite.

A convex function is always greater than or equal to its first-order Taylor expansion. A function is $\lambda$-strongly convex if it exceeds its linear approximation by a quadratic quantity: $f(y) \ge f(x) + (y-x)\nabla f(x) + \frac{\lambda}{2} \|y-x\|_2^2$. An equivalent condition is that $f - \frac{\lambda}{2}\|x\|_2^2$ is convex.