convex function

A function f on a convex set is convex if, for every x,y in its domain and \theta\in[0,1], f(\theta x+(1-\theta)y) \le \theta f(x)+(1-\theta)f(y) — the value along any chord never dips below the graph itself. Equivalently (see first-order-convexity-condition, second-order-convexity-condition), a differentiable convex function's graph never lies below any tangent line, and a twice-differentiable one has a positive semidefinite Hessian everywhere.