Strictly Convex Function

Last updated Nov 1, 2022

Let $V$ be a Vector Space over $\mathbb{R}$, let $S \subset V$ be a Convex Set, and suppose $f: S \to \mathbb{R}$. Then $f$ is a Strictly Convex Function if for all $\lambda \in (0, 1)$ and all $x,y \in S$ so $x \neq y$ $$f(\lambda x + (1-\lambda) y) < \lambda f(x) + (1- \lambda) f(y)$$

1. A Strictly Convex Function is a Convex Function.
2. What about Equivalent Conditions for Convexity? Do they have analogies for Strictly Convex Function? TODO