For all ${ 0\leq t \leq 1}$ and all ${ x_{1},x_{2}\in X,}$ ${ f\left(tx_{1}+(1-t)x_{2}\right)~\leq ~tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right).}$

Quadratic form in variables ${ x_1,x_2…, x_n}$ is a polynomial function $Q$, where all the terms in $Q(x_1, x_2,…, x_n)$ have order two. Quadratic functions $\neq$ convex functions.

Let $Q(x_1, x_2,…, x_n)=\boldsymbol{x}^TA\boldsymbol{x}$ be a quadratic form in n variables, $A$ is an sysmetric matrix, then:

- $Q$ is convex/concave $\iff$ $A$ is positive/negative semidefinite
- $Q$ is strictly convex/concave $\iff$ $A$ is positive/negative definite