I am trying to show the equivalence of two definitions of strictly convex functions. Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function.
The function $f$ is strictly convex if for each $p\in\mathbb{R}^n$ we have that (Hess$f$)$_p:=\left[\left.\frac{\partial^2 f}{\partial x^i\partial x^j}\right|_p\right]$ is a positive definite matrix (i.e. is such that $u^T(\text{Hess}f)_pu>0$ for all $u\in\mathbb{R}^n\backslash\{0\}$).
The function $f$ is strictly convex if for all $u,v\in\mathbb{R}^n$ and $0< t< 1$ we have that $f((1-t)u+tv)<(1-t)f(u)+tf(v)$.
The reason I am doing this is because I am trying to show that if a strictly convex function has a local minimum then it is in fact a global minimum. I can prove this using the second bullet, but the definition I am working with is the first bullet.
To prove the equivalence I have tried using Taylor's theorem, but I can't get it to work out nicely.
I have also tried using bullet 1 to prove the a local minimum is a global minimum directly, but cant get that to work either.