Wiki gives equivalent definitions of strong convexity here : https://en.wikipedia.org/wiki/Convex_function#Strongly_convex_functions
That is $f:\mathbb{R}^d\to \mathbb{R}$ is strongly convex if :
1) $$(x-y)\cdot (\nabla f(x)-\nabla f(y))\geq m||x-y||^2$$ and
2) $$f(y)\geq f(x)+\nabla f(x)\cdot(y-x)+\frac{m}{2}||x-y||^2$$
Does anyone know a proof of this equivalence? Confused where division by $2$ will come from.