I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$ such that $\forall x,y\in\mathbb{R}^n$ I have that $f(x)\ge\cfrac{1}{2}f(x-y)+\cfrac{1}{2}f(x+y)$.
How can I prove that $\forall x,y\in\mathbb{R}^n$ and $\forall t\in[0,1]$ I have that $f(tx+(1-t)y)\ge tf(x)+(1-t)f(y)$ ?
Firstly, $f(x)\ge\frac{1}{2}f(x-y)+\frac{1}{2}f(x+y)$ is no different from $f(\frac{x+y}{2})\ge\frac{1}{2}f(x)+\frac{1}{2}f(y)$.
Secondly, Midpoint-Convex and Continuous Implies Convex
Thirdly, A Lebesgue measurable function on an interval C is concave if and only if it is midpoint concave: https://en.wikipedia.org/wiki/Concave_function