Definition (891)

The function $\varphi:(a,b)\to \mathbb{R}$ is convex on $(a,b)$ if and only if: $\forall {0}\le \lambda\le 1,\forall a<x,y\le b$: $$\varphi(\lambda x+(1-\lambda)y)\le \lambda \varphi (x)+(1-\lambda)\varphi(y)$$ equivalently $\forall a<s<t<u<b$: $$\frac{\varphi(t)-\varphi(s)}{t-s}\le \frac{\varphi(u)-\varphi(t)}{u-t}$$ # Definition (474) The function $f:K\to\mathbb{R}$, where $K$ is a convex subset of $\mathbb{R}^n$, is called convex on $K$ if $\forall x_1,x_2\in K$ and $\lambda\in[0,1]$ $$f(\lambda\vec x_1+(1-\lambda)\vec x_2)\le\lambda f(\vec x_1)+(1-\lambda)f(\vec x_2)$$ Also if strict inequality holds whenever $\vec x_1\not=\vec x_2$ and $0<\lambda<1$, then $f$ is called strictly convex.

Proposition (Properties of Convex Functions)

1. Continuity: Convex functions are continuous on $K$ (except possibly the boundary of $K$)
2. Monotonicity of derivative equivalent to convexity: $$K\subset\mathbb{R}, f\in C^2\implies(f \text{ convex}\iff f''\ge0)$$
3. If $K\subset\mathbb{R}^n$ and $f$ is twice-differentiable in all its variables, then $f$ is convex if and only if it’s Hessian $\nabla^2 f=\left[\frac{\delta^2f}{\delta x_i\delta x_j}\right]_{i,j}$ is positive semidefinite.