Strictly convex function

Let $X$ be a vector space and let $f:X\to\mathbb{R}$ be extended-real-valued. The function $f$ is strictly convex if for all $x,y\in \mathrm{dom}(f)$ with $x\neq y$ and all $\lambda\in(0,1)$,

Context. Strict convexity strengthens convexity and typically yields uniqueness of minimizers in optimization problems.

Examples:

• On $\mathbb{R}$, $f(x)=x^2$ is strictly convex.
• On a normed space, $f(x)=\|x\|^2$ is strictly convex in many settings (e.g., Hilbert spaces).
• $f(x)=|x|$ on $\mathbb{R}$ is convex but not strictly convex (equality holds on rays with the same sign).