Strictly convex function

Let $X$ be a real vector space and let $f:X\to(-\infty,+\infty]$ have a convex effective domain

\operatorname{dom}(f)=\{x\in X:f(x)<+\infty\}.

The function $f$ is strictly convex if, for all distinct $x,y\in\operatorname{dom}(f)$ and all $\lambda\in(0,1)$ ,

f(\lambda x+(1-\lambda)y)<\lambda f(x)+(1-\lambda)f(y).

Examples

On $\mathbb R$ , $f(x)=x^2$ is strictly convex.
On a real Hilbert space, $f(x)=\lVert x\rVert^2$ is strictly convex.
The function $f(x)=|x|$ on $\mathbb R$ is convex but not strictly convex: equality holds between distinct points on the same ray.

Remarks

Strict convexity strengthens convexity. If a strictly convex function attains a minimum on a convex set, that minimizer is unique.