Show convexity of the quadratic function

Question

Given symmetric positive semidefinite matrix $A$, let

$$F(x) := x^TAx + b^Tx + c$$

Can someone show that $F$ is convex using the definition (without taking the gradient)?

Answer 1

By definition of convex, for any $x,y\in\mathbb R$, we have $$f(\frac{x+y}2)\leq\frac12(f(x)+f(y))$$ Thus it is sufficient to reduce and prove that $$\frac12(x+y)^TA(x+y)\leq x^TAx+y^TAy\\ x^TAy+y^TAx\leq x^TAx+y^TAy$$ Namely $$(x-y)^TA(x-y)\geq0$$ which is directly followed by positive semi-definite.