Real Strictly Convex Functions

Why

We want a condition for a unique minimizer.

Suppose $X \subset \R $ is convex. A function $f: X \to \R $ is strictly convex if

\[ f(tx + (1-t)y) < tf(x) + (1-t)f(y) \]

for all $t \in [0,1]$ and $x,y \in X$. $f$ is strictly concave if $-f$ is convex.