A Convex Function Bounded Above on a Set Symmetric about a Point is Bounded Below on It

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $C$ be a convex subset of Euclidean space $\mathbb{R}^{n}$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $u:C\to\mathbb{R}$ be convex on $C$.

Let $D\subseteq C$ and let $x_{0}\in D$ be such that

$$x_{0}+(x_{0}-x)\in D\qquad\text{for every }x\in D,$$

and let $M\in\mathbb{R}$ satisfy $u(w)\le M$ for every $w\in D$. Write $2=1+1$.

Then

$$2\,u(x_{0})-M\le u(x)\qquad\text{for every }x\in D.$$